Geosystem modeling with Markov chains and simulated annealing

University of Calgary

PRISM: University of Calgary's Digital Repository

Graduate Studies Legacy Theses

1997

Geosystem modeling with Markov chains and

simulated annealing

Parks, Kevin

Parks, K. (1997). Geosystem modeling with Markov chains and simulated annealing (Unpublished

doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/21777

http://hdl.handle.net/1880/26779

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THE UNIVERSITY OF CALGARY

Geosystem Modeling

with Markov Chains

and Simulated Annealing

by

Kevin Parks

A DISSERTATION

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF GEOLOGY AND GEOPHYSICS

CALGARY, ALBERTA

DECEMBER, 1997

©Kevin Parks 1997

The author of this thesis has granted the University of Calgary a non-exclusive license to reproduce and distribute copies of this thesis to users of the University of Calgary Archives. Copyright remains with the author. Theses and dissertations available in the University of Calgary Institutional Repository are solely for the purpose of private study and research. They may not be copied or reproduced, except as permitted by copyright laws, without written authority of the copyright owner. Any commercial use or publication is strictly prohibited. The original Partial Copyright License attesting to these terms and signed by the author of this thesis may be found in the original print version of the thesis, held by the University of Calgary Archives. The thesis approval page signed by the examining committee may also be found in the original print version of the thesis held in the University of Calgary Archives. Please contact the University of Calgary Archives for further information, E-mail: [email protected]: (403) 220-7271 Website: http://www.ucalgary.ca/archives/

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Abstract

The objective of this dissertation is to determine if Markov statistical structures can be

imposed on structured random grids of hydraulic conductivity (K) in an effort to inject

more geological realism into stochastic simulations of aquifer heterogeneity. Cyclicity is

imposed on unconditional, continuous K fields through use of the hole-effect covariance

structure. The main effect of using the hole effect is a significant reduction in the variance

of outputs of stochastic experiments. Percolation experiments suggest that enforcement

of a vertical hole-effect covariance structure increases the probability that high values of K

are connected in the horizontal. Markov transition probability matrices are encoded into

multi-point histograms and then 2D categorical fields are constructed with simulated

annealing. Markovian structures with a geological significance are imposed on these

fields: hierarchical stratigraphic memory (double dependency), directionality, and cyclicity.

There is an effect on flow behaviour when these different structures are imposed, though

these are not first-order effects. The transference of variability information from the

vertical to the horizontal under Walther's Law of Facies Succession can be imposed in a

temporally-rescaled framework. As a field trial, a Markov statistical model of vertical

variability is constructed of a complex aquitard at the Gloucester waste disposal site,

Ontario. The layer is found to have a large component of random noise in its Markov

transition matrices. This noise obscures a general vertical directionality associated with

fining-upward successions. Correlations between lithotypes and laboratory-measured

hydraulic conductivity are confounded by the large dispersion of values within each

lithotype. Even though the Markov statistics are not elegant, they do conform to the

general depositional model for the aquitard layer, that being a coalescent, subaqueous,

proglacial outwash fan. Practical annealing issues like the form of the objective function,

cooling schedule, perturbation method, and scale effects pose significant but not

insurmountable obstacles to implementing the ideas put forth herein. The results of this

work do show that Markovian analysis and Markov field construction could imbue

stochastic models with more geological realism.

iii

Acknowledgements

This work has been supported through the Amoco Canada Petroleum Company Ltd.

Graduate Fellowship in Geology. Additional support has been provided by the Canada

Centre for Inland Waters, the Canada NSERC Grant No. OGPO122023, and the

Department of Geology and Geophysics, University of Calgary.

I certainly must acknowledge the support and mentoring provided by my supervisor,

Dr. Larry Bentley. He patiently let me explore more blind alleys than he maybe should

have. But he was always there to help precipitate structure from the nebulous cloud of

half-baked ideas and irreproducible experiments that sometimes accompanies a work such

as this. I would also like to thank the other members of my supervisory committee, Drs.

Terry Gordon and Fran Hein, for their comments and guidance at critical junctures of my

project. A very sincere note of gratitude goes to Dr. Allan Crowe at the National

Hydrological Research Institute at Burlington, Ontario, for his interest as well as practical

support of this project. I would also like to single out some fellow travelers in graduate

school for their companionship and camaraderie, particularly Guy Kieper, Mark Moncur,

Bill Hoyne, the "Dudes of Diagenesis" from Dr. Ian Hutcheon's group, as well as

neighbours and friends in the 400 Court at student family housing.

Graduate school for a family man cannot be seriously undertaken without the support

of his wife. Lorraine has always been with me on this journey, coaxing and encouraging

me to continue with my studies. For this I thank her most of all. I also thank my parents

and parents in-law for their support during these years. Finally, I need to thank my

children, Emilie and Samantha, for the happiness they always bring to me and how they

always made me keep things in perspective. They will not realize for a long time how

special our life was during these years, and I want to thank them in advance for being

there.

iv

Dedication

For Lorraine and Emilie

But especially for Samantha, who

wasn 't here for the last one.

Table of Contents

Approval Page ii

Abstract iii

Acknowledgements iv

Dedication v

Table of Contents vi

List of Tables be

List of Figures xi

List of Plates xiv

CHAPTER 1: INTRODUCTION 1

References to Chapter 1 7

CHAPTER 2: THE HYDROGEOLOGICAL SIGNIFICANCE OF HOLE-EFFECT

MODELS IN 2D STOCHASTIC SIMULATION 9

The Hole Effect as Covariance Model 10

Previous Work with Hole Effect Models in Geostatistical Estimation

and Simulation in Hydrogeology 13

The Hole Effect as a Data Artifact 14

The Hole Effect as a Geologic Signal 16

The Hydrogeological Significance of Hole-Effect Covariance Models

on Simulated 2D Flow and Transport 18

Experiment 1: Continuous Layer Models 18

Experiment 2: Isotropic 2D Models 21

Experiment 3: 2D Panels with Vertical Hole Effect Only 33

Discussion 40


vi

CHAPTER 3: CAPTURING CONCEPTUAL MODELS OF STRATAL

ARCHITECTURE IN SYNTHETIC AQUIFERS WITH MARKOV CHAINS AND

SIMULATED ANNEALING 51

Building Markov Fields by Simulated Annealing 52

Markov Chains and Fields 52

Building Markov Fields with Simulated Annealing 54

Exploring Aspects of Markovian Stratigraphy 60

Double Dependency in Markov Fields 61

Directionality in Markov Fields 65

Cyclicity in Markov Fields 68


CHAPTER 4: INFORMING HORIZONTAL MARKOV MEASURES OF

VARIABILITY WITH THE VERTICAL 72

Coordinate Transforms in Markov Fields and Walther's Law 73

A Demonstration 75


CHAPTER 5: MARKOV CHARACTERIZATION OF VERTICAL VARIABILITY IN

A COMPLEX AQUITARD UNIT: GLOUCESTER WASTE DISPOSAL SITE,

ONTARIO 84

General Geology of the Area of the Gloucester Waste Disposal Site 88

The 1995 Sampling and Analysis Program 90

Lithotypes 91

Markov Descriptions of Vertical Variability 94

Embedded Markov Chains 98

Homogeneity of Depositional Process 100

Conventional Markov Description 102

Length Scale Information 109

vii

Embedded Markov Chain Analysis of the Confining Layer 113

Conductivity and Porosity of the Gloucester Confining Layer 117

Hydraulic Conductivity 117

Porosity 124

Discussion 124


CHAPTER 6: THE PERFORMANCE OF SIMULATED ANNEALING WITH

RESPECT TO STOCHASTIC RECONSTRUCTION OF HETEROGENEITY FROM

MARKOV STATISTICS 133

Implementation and Form of the Objective Function 134

Effects of the Cooling Schedule 136

Sensitivity to Length Scales on Finite Grids 140

Iterative Improvement or Simulated Annealing? 143

Two Dimensional Reconstruction of the Gloucester Confining Layer

by Annealing 145


CHAPTER 7: CONCLUSION 154

COMPLETE BIBLIOGRAPHY 157

APPENDICES

Appendix ArFortran code for percolation experiments 170

Appendix B. Fortran codes for annealing Markov fields 176

Program listing for manneal. for 177

Program listing for prephist.for 225

Sample input files markov.par and anneaLpar 242

Appendix C. Core photographs, borehole logs and hydraulic data 243

viii

LIST OF TABLES

CHAPTER 3

Table 3.1: A Markov transition matrix 53

Table 3.2: A hypothetical double-dependent Markov structure 62

Table 3.3: The single dependent transition matrix embedded in Table 3.2 62

Table 3.4: A Markov transition matrix with both directionality and cyclicity 67

Table 3.5: The same matrix but with directionality removed 69

CHAPTER 5

Table 5.1: Markov transition frequency matrix for east subset of cores 102

Table 5.2: Markov transition frequency matrix for west subset of cores 102

Table 5.3: Single dependent transition frequency matrix for Gloucester Confining

Layer 105

Table 5.4: Single dependent transition probability matrix for Gloucester Confining

Layer 105

Table 5.5: Comparison of observed versus calculated proportions of lithotypes in

Gloucester Confining Layer 106

Table 5.6: Transition frequency matrix for filtered data 107

Table 5.7: Transition probability matrix for filtered data 107

Table 5.8: Hypothetical depositional rates for lithotypes 108

Table 5.9: Markov transition frequency matrix after temporal rescaling 109

Table 5.10: Markov transition probability matrix after temporal rescaling 109

Table 5.11: Comparison of marginal probability vectors from temporally rescaled and

geometric Markov transition matrices 110

Table 5.12: Comparison of mean bed thicknesses with expected values 112

Table 5.13: Comparison of mean bed thicknesses with median body influence lengths... 113

Table 5.14: Upward embedded transition matrix for Gloucester Confining Layer 114

Table 5.15: Values of Turk's test statistic 115

Table 5.16: Substitutability matix for Gloucester Confining Layer 117

ix

CHAPTER 6

Table 6.1: A simple three-state Markov transition matrix 139

Table 6.2: Four variants of the standard annealing schedule 139

Taible 6.3: Four simple Markov matrices with different body influence lengths 142

Table 6.4: A more complicated Markov transition matrix 144

Taible 6.5: Markov transition matrix for filtered Gloucester data set after combining silty

clay and clay lithotypes 150

Table 6.6: Global proportions and transition frequencies for dip section reconstruction of

Gloucester Confining Layer 151

Table 6.7: Global proportions and transition frequencies for strike section reconstruction

of Gloucester Confining Layer 153

Table 6.8: Expected mean thicknesses for transition probabilities in Table 6.5 150

APPENDIX C

Table of Measured Hydraulic Properties 290

x

LIST OF FIGURES

CHAPTER 2

Figure 2.1: Illustration of model variogram withahole effect 11

Figure 2.2: One dimensional conductivity fields and their variograms 20

Figure 2.3: Experimental variograms of cell fluxes in layered systems 22

Figure 2.4: 2D isotropic fields with a gaussian structure and a hole-effect structure and

their experimental variograms 24-26

Figure 2.5: Comparison of calculated effective conductivities 27

Figure 2.6: Illustration of extreme path value concept 29

Figure 2.7: Distributions of extreme path values 30

Figure 2.8: Maps of variance in predicted heads 32

Figure 2.9: Means and standard deviation in calculated longitudinal dispersivity 34

Figure 2.10: 2D anisotropic panels with a spherical structure and a hole-effect structure in

the vertical and their experimental variograms 36-37

Figure 2.11: Histograms of effective horizontal and vertical hydraulic conductivities and

their ratios 38

Figure 2.12: Means and standard deviation in calculated longitudinal dispersivity 39

Figure 2.13: Histograms of extreme path conductivities for 2D panels 41

CHAPTER 3

Figure 3.1: Diagramatic explanation of simulated annealing 55

Figure 3.2: Illustration of multipoint histogram concept 58

Figure 3.3: Two unconditional Markov fields, one with double dependency 63

Figure 3.4: Histograms of effective conductivities, single vs. double dependency 64

Figure 3.5: Unconditional fields with directionality and cyclicity 66

X I

CHAPTER 4

Figure 4.1: Markov transition matrices for a three-state system before and after rescaling

by depositional rates 76

Figure 4.2: A three state, isotropic Markov field in 2D space coordinates 79

Figure 4.3: A three state, isotropic Markov field in time-space coordinates 80

Figure 4.4: The three state, isotropic Markov field backtransformed to space 2D

coordinates 81

CHAPTER 5

Figure 5.1: Location of Gloucester waste disposal site 85

Figure 5.2: Geology of Gloucester waste disposal site 86

Figure 5.3: Contaminant plume at Gloucester waste disposal site 87

Figure 5.4: Map of borehole locations 92

Figure 5.5: West-east cross-section X-Y 95

Figure 5.6: North-south cross-section Y-Z 96

Figure 5.7: Thickness distribution of beds by lithotype 110

Figure 5.8: Scatter plot of no-load versus loaded Kv 118

Figure 5.9: Histogram of all no-load Kv measurements 120

Figure 5.10: Histograms of Kv by lithotype 121

Figure 5.11: Vertical variogram of logioKv 122

Figure 5.12: Horizontal variograms of logioKv 123

Figure 5.13: Histograms of porosity by lithotype 125

Figure 5.14: Scatterplot of logio Kv versus porosity 126

CHAPTER 6

Figure 6.1: Comparison of objective function behaviour with different annealing

schedules 139

Figure 6.2: Effect of length scale on annealing performance 142

xii

xiii

Figure 6.3: Comparison of objective function trajectories for true annealing, iterative

improvement, and combination 144

Figure 6.4: Isotropic Markov field representing a dip section through the Gloucester

Confining Layer 146

Figure 6.5: Isotropic Markov field representing a strike section through the Gloucester

Confining Layer 147

APPENDIX C

Borehole log legend 250

Log of Borehole UC95-2 251














xiii

List of Plates

APPENDIX C

Plate C.l: Representative photograph of medium-coarse sand lithotype 244

Plate C.2: Representative photograph of fine sand lithotype 245

Plate C.3: Representative photograph of silty lithotype 246

Plate C.4: Representative photograph of silty clay lithotype 247

Plate C.5: Representative photograph of clay lithotype 248

Plate C.6: Representative photograph of diamict lithotype 249

xiv

1

Chapter 1

Introduction

Predicting hydraulic head and solute concentration in groundwater at unsampled

locations in aquifers is a primary goal of hydrogeologic investigations. Since sampling is

expensive, hydrogeologists rely on predictive equations and models to estimate changes in

groundwater flow and chemistry between sample points and over time. The results are

used to assess the need to remediate subsurface contamination or to justify a no-action

alternative within a risk assessment framework. Variants of the same tools are used in

petroleum-reservoir engineering.

Because aquifers are never exhaustively sampled by boreholes, there is always

uncertainty in the values of hydraulic parameters used in these equations and models. To

assess the impact of this uncertainty on predicted behaviour, three main approaches have

evolved: stochastic simulation, inverse techniques, and stochastic differential equations. In

the first approach, uncertainty in prediction is assessed by using multiple, equiprobable

realizations of aquifer heterogeneity to populate flow-simulator grids. Multiple runs of the

flow simulator allow exploration of the predictive uncertainty of flow behaviour

concomitant with the underlying parameterization uncertainty. Geostatistical or

optimization methods are used to generate these so-called stochastic realizations. The use

of multiple realizations to assess uncertainty through numerical calculation is often referred

to as a Monte Carlo experiment or stochastic simulation.

The second approach refers to the coupled use of hydraulic head data and flow

simulation to derive an optimal set of model parameters. The third approach makes use of

analytical techniques to solve the underlying differential equations directly with embedded

error terms. The latter two approaches are not discussed in this dissertation. Koltermann

and Gorelick (1996) review the current state-of-the-art of geosystem modeling and grid

parameterization techniques.

2

Deutsch and Hewitt (1996) list a number of challenges in forecasting petroleum

reservoir performance that apply in equal measure to predicting solute transport in

aquifers. Amongst these challenges are 1) the identification of geologic features that have

a first-order impact on flow and transport behaviour and 2) capturing them in a numerical

or statistical descriptor suitable for informing the distribution of hydraulic conductivity (K)

on a flow simulator grid. Statistics like the mean, variance, and spatial covariance of K are

well known to have a first-order impact on flow behaviour. But these descriptors are not

first-order controls on transport behaviour because they are insensitive to the geometry

and continuity of extreme values of K. Connected extreme values can be channels or

barriers to flow.

Doveton (1994) suggested that Markov geosystem models can play a role in

transmitting more geological realism into flow simulator grids. Markov statistics

encapsulate information on relationships between categories as well as length-scale

information. In their most common form, Markov statistics are presented in form of a

transition probability matrix. The matrix tabulates the probability that a time or space

series stays in the same state or enters a different state with each succeeding step. Markov

structures have long been used by geologists to identify and quantify facies relationships in

bedding sequences (e.g., Schwarzacher, 1975; Walker, 1979). Various methods can be

used to generate Markov fields directly. One can use also Markov information to modify

other geostatistical constructs (see Chapter 3).

Despite their obvious attraction to geologists, Markov fields have received relatively

little attention by geosystem modellers involved in flow simulation Koltermann and

Gorelick (1996) give Markov fields but a few paragraphs in their comprehensive review.

They cite the difficulty of conditioning Markov fields to field data as a barrier to their

practical use. They also question the dubious (at least to geostatisticians) suggestion that

Markov temporal signatures from one-dimensional sequences can be extended to

multidimensional spatial grids in a meaningful way.

3

In this dissertation, I explore the use of Markov structures in geosystem modeling for

hydrogeologic simulation. In particular I ask: do geologically meaningful Markov

structures have an effect on flow and transport behaviours in flow simulators? If so, then

these structures may have application in geosystem modeling where complex stratal

relationships are important controls on flow and transport. To the best of my knowledge,

no one has systematically explored this question before.

To do this work, I used simulated annealing to build both continuous and categorical

fields with Markov structures. Simulated annealing is a global optimization technique

recently adapted to generate stochastic images of petroleum reservoirs conditioned to

honour data from disparate sources. For experiments involving continuous fields, I

adapted a GSLIB annealing program published by Deutsch and Journel (1992). For

categorical fields, I wrote my own annealing code. In the course of this work I

encountered some important performance issues that concern the generation of Markov

fields by this method. These observations will be of interest to those concerned with the

art of stochastic image generation by simulated annealing. Flow and transport experiments

were conducted with the USGS finite-difference code Modflow (McDonald and

Harbaugh, 1988) and the particle tracking code Modpath (Pollock, 1989). Percolation

threshold experiments were conducted with my own code.

As the field component of this work, I attempted to build a two-dimensional stochastic

representation of a complex, heterogeneous aquitard unit using Markov structures and

simulated annealing,. The Markov structures were built from geologic descriptions and K

values garnered from conventional split-spoon cores. The cores were collected by me

with the generous assistance of Environment Canada in 1995. This novel application will

be of interest to hydrogeologists keen on applying stochastic simulation techniques to

populate flow model grids but possessing only descriptive geologic information from

regional survey reports and low-cost, low-tech sources like split-spoon cores, trenches,

grain-size analyses, and falling head permeameters. I foresee that this work can be further

4

extended within a Bayesian framework for site investigations (e.g., Rosen and Gustafson,

1996) but this topic is beyond the scope of this study.

The contents of this dissertation are organized as follows:

In Chapter 2, the hydrogeological significance of using a particular covariance

structure called the "hole-effect model" in stochastic simulation of continuous K fields is

considered. Often ignored as a data artifact, the hole effect can be the Markovian

signature of allocyclic or autocyclic controls in sedimentary depositional systems.

Numerical experiments suggest that selection of this kind of covariance structure in

stochastic realizations, whether warranted by geology or not, affects the output of flow

and transport simulation. Guidelines for selection of covariance models based on geologic

interpretation are suggested.

In Chapter 3, geologic complexity is introduced into categorical fields through

Markov transition matrices. Markov statistics have long been known to capture

stratigraphic architecture which can be related to sedimentary process. While

one-dimensional simulation of Markov fields is trivial, simulation in higher dimensions is

more difficult. Here, two dimensional Markov fields are built by simulated annealing with

multipoint histograms. Numerical experiments show how various Markov stratigraphic

architectures can be stochastically generated. Flow properties of the fields are shown to

be sensitive to Markovian stratigraphic architecture.

One-dimensional Markov statistics from the vertical are often assumed to be

traiismittable to the horizontal through Walther's Law of Facies Succession. In Chapter 4,

the time-stratigraphy implications of this assumption in Markov-field simulation are

considered.

5

Chapters 5 and 6 document my attempt to use a Markov-field methodology to

reconstruct complex aquitard stratigraphy at the Gloucester Landfill site near Ottawa.

This famous site was the subject of many pioneering attempts at delineating and

remediating a dissolved contaminant plume (e.g., Jackson et al., 1991; Gailey and

Gorelick, 1993). Detailed sedimentological studies of the complex sediments hosting the

plume were made by others in the 1980s. Unfortunately, the qualitative techniques used to

populate simulator grids in that era were such that geologic conceptual models were of

little use when attempting to define aquifer heterogeneity at the interwell scale. Instead

lithostratigraphic correlation prevailed and subsequent site models were based on a layered

system. The layers comprised a surficial unconfined sand aquifer, the senuconfining silty

clay aquitard mentioned above, and a thick, semiconfined sand and gravel aquifer

overlying bedrock.

Of interest to my study is whether I could capture some essence of complex

interbedding in the silty clay aquitard with Markov structures and then recreate them in a

gridded model. New sample cores from this interbedded layer were collected in October

1995 with the assistance of Environment Canada staff and their hollowstem drilling rig.

The cores were characterized in terms of lithology, vertical hydraulic conductivity, and

porosity. Markov and conventional geostatistical measurements of the cores and their

propenies were made. This descriptive work constitutes Chapter 5.

Chapter 6 demonstrates the combination of the descriptors and statistics reported in

Chapter 5 with the concepts of Chapter 3 and 4 to reconstruct complex aquitard

stratigraphy of the Gloucester site. This effort meets with mixed results, in part due to the

large random noise component in the bedding relationships. It is ultimately concluded

that a simpler geologic environment may be more amenable to this approach. Performance

issues related to this approach to transmitting geologic knowledge into stochastic fields via

a Markov formulation and simulated annealing are documented here.

The results are summarized in Chapter

documentation of programs developed in

investigation at Gloucester.

6

7, the Conclusion. The appendices include

this work as well as field data from the

7

References to Chapter 1

Deutsch, C.V., and T.A. Hewitt, 1996. Challenges in reservoir forecasting. Mathematical

Geology, vol. 28, no. 7, p. 829-842.

Deutsch C.V. and A.G. Journel, 1992. GSLIB Geo statistical Software Library and User's

Guide. Oxford University Press, New York. 340 pp.

Doveton, J.H., 1994. Theory and applications of vertical variability measures from

Markov chain analysis. In: Yarus, J.M., and R.L. Chambers, eds. Stochastic Modeling

and Geostatistics - Principles, Methods, and Case Studies. American Association of

Petroleum Geologists Computer Applications in Geology, no.3. AAPG, Tulsa,

Oklahoma, p. 55-64.

Gailey, R.M., and S.M. Gorelick, 1993. Design of optimal, reliable plume capture

schemes: application to the Gloucester Landfill ground-water contamination problem.

Ground Water, vol. 31, no. 1, p. 107-114.

Jackson, R.E., S. Lesage, M.W. Priddle, A.S. Crowe, and S. Shikaze, 1991. Contaminant

Hydrogeology of Toxic Organic Chemicals at a Disposal Site, Gloucester, Ontario. 2.

Remedial Investigation. Inland Waters Directorate Scientific Series No. 181. National

Water Research Institute, Environment Canada, Burlington, Ontario. 68 pp.

Koltermann, C.E., and S.M. Gorelick, 1996. Heterogeneity in sedimentary deposits: A

review of structure-imitating, process-imitating, and descriptive approaches. Water

Resources Research, vol. 32, no. 9, p. 2617-2658.

McDonald, M.G., and A.W. Harbaugh, 1988. MODFLOW: A Modular

Three-Dimensional Finite Difference Ground-Water Flow Model. Techniques of Water-

8

Resources Investigations of the United States Geological Survey, U.S. Geological Survey,

U.S. Department of the Interior.

Pollock, D.W., 1989. Documentation of computer programs to complete and display

pathlines using results from the U.S. Geological Survey modular three-dimensional

finite-difference ground-water model. U.S.G.S. Open File Report 89-381, 81 pp.

Rosen, L., and G. Gustafson, 1996. A Bayesian-Markov geostatistical model for

estimation of hydrogeological properties. Ground Water, vol. 34, no. 5, p. 865-875.

Schwarzacher, W., 1975. Sedimentation Models and Quantitative Stratigraphy.

Developments in Sedimentology 19, Elsevier, 382 pp.

Walker, R.G., 1979. Facies and Facies Model. General Introduction. In: R.G. Walker,

ed., Facies Models, 1st Edition. Geoscience Canada Reprints Series 1. p. 1-8.

9

Chapter 2

The Hydrogeologic Significance of Hole-Effect Models in 2D Stochastic Simulations.

Stochastic simulation is a family of techniques for populating model grids with

hydraulic parameters (e.g., hydraulic conductivity (K), porosity) at unsampled locations.

At the heart of each technique lies a statistical descriptor which captures some essence of

the heterogeneity present in the real system Stochastic simulators assign parameters to

grid blocks in a way that reproduces the statistical descriptor of the real system. Because

the descriptor is statistical, there may be an infinite number of equiprobable, unique

combinations of grid-block values capable of matching the statistical descriptor. Each

equiprobable combination is termed a realization. If the simulator technique enforces

measured values of parameters at grid blocks corresponding to sampled points in each

realization, the technique is said to be conditional. Otherwise the technique is

unconditional.

The most commonly used geostatistical descriptor of geologic heterogeneity is the

two-point covariance or variogram. This descriptor captures the decay of correlation in

parameter values with increasing distance between any two points in a field (see Issaks and

Srivistava, 1989). With most standard covariance models, there will be a distance called

the range beyond which point values are uncorrelated. The covariance of points equals

the sample population variance at a lag of 0 and decreases to zero at the range.

On a regular grid of points, the two-point covariance is calculated as:

a*) = TA) ifflMxym,) (z(X+h)-m,+„)] (2.1)

10

where C(h) is the two-point covariance for the separation or lag vector, h, n(h) is the

number of points separated by the vector h, z(x) is the parameter value at location x, m, is

the mean value of the n points at x, z(x+h) is the value of the point displaced by the lag

vector b away from z(x), and mx+h is the mean of those values.

For historical reasons, the variogram or semivariance is usually calculated instead of

the covariance:

Y(h) = 2**) 22? [(Z(x)-z(x+h)]2 (2.2)

The semivariance increases from zero at a lag of 0 and rises to a value equal to the

sample population variance, also termed the sill of the variogram, at lags beyond the

range. Srivastava and Parker (1989) and Issaks and Srivistava (1989) explore the

properties of the covariance, semivariance and related measures. Deutsch and Journel

(1992) provide algorithms for building these descriptors from irregularly spaced data using

angular and length tolerances on the lag vector.

The Hole Effect as Covariance Model

A variogram whose growth to a sill within a finite range is not montonic is said to

possess a "hole effect" (Journel and Huijbregts, 1978, p. 168) (Figure 2.1). Hole effects

are commonly observed in experimental variograms of geologic media, especially in the

vertical (e.g., Aasum and Kelkar., 1991, their figure 2; Desbarats and Bachu, 1994, their

figure 5; Kittridge et al., 1990, their figure 8; Grant et al., 1994, their figure 12). A hole

effect in a variogram suggests a renewed improvement in statistical correlation at some lag

(and its multiples), interrupting or even reversing the general decay with distance of the

correlation of parameter values between neighbouring points.

11

o

s >

0>

in

Lag

Figure 2.1: Illustration of a model variogram with a hole effect (solid line) as compared to a model variogram with a monotonically increasing structure (dashed line).

12

Review of the literature shows that hole effects in experimental variograms are often

ignored when choosing a covariance model to enforce in stochastic simulation for flow

models. This choice is justified if the hole effect is a data artifact. But if the hole effect is

a real attribute of geologic heterogeneity, then we must ask: what is the hydrogeologic

significance of ignoring this extra information when using stochastic simulation techniques

to model real geologic systems? Similarly, what would be the impact if a false hole effect

is enforced? Can knowledge of geologic systems which produce hole-effect covariance

structures be of value when choosing a covariance structure for simulation when data are

sparse?

The specific objective of this chapter is to examine the flow and transport effects of

incorporating a hole-effect model of covariance in 2D stochastic representations of

heterogeneous porous media. The more general question, pertinent to the theme of this

dissertation, is to question whether covariance structures are an efficient vehicle to

transmit geological information into stochastic simulations.

Experiments show that the major contribution of incorporating a hole effect is a

statistically significant reduction in variance of flow and transport behaviour. There is also

evidence that when a vertical hole-effect model is enforced in weakly anisotropic 2D

conductivity fields, there may be better connectivity of extreme values in the horizontal.

Otherwise, there is no first-order difference in the flow and transport properties of the

fields. This conclusion is significant to this disseration in two ways:

1. The reduction in variance shows that the extra effort to incorporate the geologic

knowledge in the form of a hole effect reduces the space of uncertainty explored by a

family of realizations.

2. At the same time, the lack of significant difference in flow properties of random

fields differing only by two-point covariance structure underscores that this

13

popular statistical descriptor has limited ability to capture heterogeneity of real geologic

systems.

This chapter proceeds by examining the hole effect first as a geologic signal, then as a

data artifact. Following that discussion, experiments are presented which support the

conclusions mentioned above.

Previous Work with Hole Effect Models in Geostatistical Estimation and

Simulation in Hydrogeology

Hole-effect models in geostatistical estimation (as opposed to simulation) are

described in Journel and Huijbregts (1978). Journel and Froidevaux (1982) used a

combined directional hole-effect model with a nested spherical model to krig ore grades in

a tungsten deposit that showed a strong directional periodicity due to folding. Gelhar

(1986) reported that predicted head variances from stochastic equations of flow will be

veiy sensitive to the form of the assumed covariance function. He called for further

research into methods for identifying appropriate input covariance structures from

geologic knowledge.

Johnson and Driess (1989) noted that hole effects are much more common in the

vertical than the horizontal because cyclic or pseudo-periodic repetition of layers with

similar properties is more common than regular horizontal patterns of lenses. Their field

study did not show any hole effects that could be correlated to geology. But, significantly

for this study, their work demonstrated that experimental variograms are very sensitive to

radial and angular (dip) tolerances in searches on irregular data patterns. This sensitivity

means that complex geological signatures expressible in a covariance measure can be

easily missed in experimental variograms constructed with the radial and angular

tolerances typically used in real data sets.

14

Fogg (1989) produced 3D stochastic simulations of sand bodies with a hole effect in

the horizontal as well as the vertical. He compared the effective flow properties with

those simulations generated with a monotonically smooth spherical model. His results

indicated that the simulated sand-body distributions made with the hole effect model were

more compartmentalized with regard to his measures of sand-body interconnectedness. He

also noted that this compartmentalization had little impact on effective flow properties.

He surmised that these variations in connectivity would have an impact on solute

transport, but this was not measured in his study.

Ouenes and Bhagavan (1994) reported an experiment wherein an exhaustive data set

exhibiting a strong hole effect in the vertical and a simpler structure in the horizontal was

better reproduced by simulated annealing when none or only part of the vertical hole

structure was honoured than when an incorrect vertical model was used. Sen et al. (1995)

showed a satisfactory history match of simulated tracer effluent when an

outcrop-measured exhaustive experimental variogram with a hole effect was honoured in

stochastic simulation.

Jensen et al. (1996) considered geological meaning of hole effects in semivariograms

in some detail, but they did not specifically explore its control on flow and transport in

groundwater simulations.

The Hole Effect as a Data Artifact

Hole effects in experimental variograms often have dubious physical meaning because

they can be produced by a number of data artifacts (Journel and Froidevaux, 1982).

Simply extending lags across more than one-half of a sample domain can introduce an

artificial hole effect. Outlier high and low values in sparse data sets can likewise introduce

oscillations in variograms. Pairing of low values with high values will produce high

variogram values. Likewise, pairings of low-low and high-high

15

values will produce low variogram values. In sparse data sets there may not be sufficient

other data to dampen their contributions to the variogram calculation. Journel and

Huijbregts (1978, p. 247) showed how an observed hole effect in a single borehole

disappeared when the data were combined with data from adjacent boreholes. Similarly,

Schwarzacher (1975) discussed how erratic high values can produce artificial oscillations

in experimental autocorrelation functions produced from short observational series.

Armstrong (1982) showed how extreme values, intermingled populations, and simple user

errors can introduce oscillatory behaviour in experimental variograms. Bayer (1985)

showed how the introduction of small random fluctuations in a regular sampling interval

on an idealized periodic function can produce pseudo-periodic oscillations of higher

frequency in a variogram or autocorrelation function.

So why even concern oneself with hole effects? Journel and Froidevaux (1982) opined

that if independent geologic evidence suggests a reasonable physical basis for true or

pseudo-periodic behaviour in a medium, then a hole-effect model should be used because

it brings in more information on the spatial geometry of heterogeneity. Still, one must be

cautioned against using subjective geologic observation of cyclicity in bedding sequences

as solid evidence for incorporating a hole effect in a covariance model. In a famous study,

Zeller (1964) demonstrated how geologic interpretation can easily introduce cyclcity

where none exists just because the human mind is predisposed to look for patterns.

(Zeller had geologists correlate sequences of numbers as beds which originated as

numbers chosen at random from a telephone directory.)

The Hole Effect as a Geologic Signal

The occurrence of hole effects in covariance measures of sedimentary rock may be due

to regular or pseudo-cyclicity in bedding. Cyclicity implies the regular recurence of a bed

or a pattern of beds. Regular spatial repetition of like values in cyclic bedding sequences

will create high values of covariance (low semivariance) at lags comparable to the vertical

16

distance between repeated beds. Similarly, regular pairing of unlike values will create low

or even negative covariance values. On a variogram, cyclic pairing of like and unlike beds

will create oscillations of the semivariogram about the sill.

The notion of cyclicity in bedding is deeply embedded in stratigraphy and

sedimentology. A large literature exists on cyclicity in sedimentary rock at various scales.

This discussion is not presented as an exhaustive review but to serve as a reminder of the

multitude of geologic processes that can generate a cyclical stratal architecture at all

scales.

Cyclicity or periodicity can be defined as a group of different lithologies that reoccur

with some regularity or pattern in a geologic sequence (Schwarzacher, 1993) . In terms of

a covariance structure, true or mathematical periodicity will appear as oscillations which

maintain their amplitude with increasing lag. Pseudo-periodicity, on the other hand,

manifests itself as a hole effect that dampens to zero with increasing lag. A power

spectrum of a truly periodic process will have spikes at dominant frequencies whereas a

pseudo-periodic series will have a power spectrum dominated by low frequencies (Box

and Jenkins, 1976).

In sediments, true periodicity can occur when depositional processes are strongly

controlled by an external mechanism with a periodic oscillation. Such depositional

processes are termed allocyclic (Miall, 1980). True mathematical periodicity can

conclusively be shown in processess directly linked to extraterrestrial orbital forcings

(Fischer and Bottjer, 1991; Schwarzacher, 1993). Periodic or pseudo-periodic

sedimentological response to daily (solar) and yearly (calendar) forcings are well

documented. Deposits with demonstrable external forcings include: diurnal

sedimentological reorganization in glacier-fed streams (Hein and Walker, 1977); tidal flow

reversals (Fischer and Bottjer, 1991); glaciolacustrine varves representing seasonal control

17

on sedimentation (Agterberg and Banerjee, 1969); limestone-marl couplets recording

seasonal variations in algal productivity in lakes (Fischer and Roberts, 1991).

On a longer time-scale, sedimentary systems show allocyclic control in the so-called

Milankovitch band. Milankovitch-band cycles include climatic variations ascribed to cyclic

variations in the precession of the earth's rotational axis as well as the obliquity and

eccentricity of the earth's orbit around the sun. Because of the nearly perfect

mathematical regularity of the earth's orbital and rotational variations, the sedimentary

record of processes controlled by Milankovitch cycles may show true periodicity on time

scales of 95 ka to 2 Ma. Some of the geological responses argued to be allocyclically

controlled by Milankovitch processes include: glaciation (summarized in Fischer and

Botjjer, 1991 and Schwarzacher, 1993); fluvial deposition in response to glacioclimatic

fluctuations (e.g. Ashley and Hamilton, 1993); glacio-eustatic global sea level changes

(Vail et al., 1991); changes of river base levels and regional water-table fluctuations tied to

glacio-eustatasy (Shanley and McCabe, 1994); and climatic controls on global productivity

of algae, plankton, and bioturbation (Fischer and Bottjer, 1991).

Sedimentary response to tectono-eustasy and long wavelength periodic global sea level

changes (Vail et al., 1991) may be recorded in the pseudo-cyclic record of parasequence

architecture on continental shelves. The driving mechanisms for these cycles are not

known. They may be related to variations in the rate of sea-floor spreading, linked to a

poorly understood mantle convection system. Higher order cycles may be related to

continental drift, global tectonics, and crustal plate configuration. It is still debatable

whether these global-scale strata! patterns are allocyclically controlled or just represent

repeated events (Wilgus et al., 1988; Einsele et al, 1991).

Cyclic bedding can also be produced by internal oscillations of the distribution of

energy in a depositional system. Such bedding patterns are called autocyclic (Miall,

1980). They are initiated by external disturbances to dynamic equilibria of sedimentary

18

systems or by the existence of feedback mechanisms. Oertel and Walton (1967)

introduced the concept of feedback mechanisms in deltaic deposits, showing how cyclic

bedding can be formed. Harbaugh and Bonham-Carter (1970) and Schwarzacher (1975)

also explored the role of feedback mechanisms in generation of autocyclicity in sediments.

MiaU (1980) discussed autocyclicity in fluvial systems and the creation of stacked channel

sequences. Power and Walker (1996) show cyclic vertical variation in prograding shelf

complexes in the Belly River Formation of Alberta and argue against an autocyclic

mechanism. Einsele et al. (1991) reported that autocyclic beds usually show only limited

stratigraphic continuity. Pseudoperiodicity in sediments has also been linked to external or

large-scale autocyclic forcings like tectonic pulses of uplift and subsidence controlling

distribution of grain sizes in alluvial fans (Miall, 1980; Neton et al, 1994).

Sharp (1982a,b) and Hohn (1988) discuss in more detail the mathematical

relationships between autocyclicity and damped oscillatory (hole effect) covariance

structures in time and space series.

The Hydrogeological Significance of Hole-Effect Covariance Models on

Simulated 2D Flow and Transport

Three experiments were conducted to examine the flow and transport effects of a

hole-effect covariance structure in 2D random fields meant to represent aquifers.

Experiment 1: Continuous Layer Models

Three layered models were produced. One model had a random vertical structure, the

second family had a smooth transitory covariance structure in the vertical, and the third

had a damped hole-effect covariance structure with an initial amplitude (height of the first

maximum oscillation above the variogram sill divided by the variogram sill) of 0.50. The

19

seniivariogram model used for the hole effect in this study is a damped cardinal cosine

model (Hohn, 1988):

y(h) = 1 - [expi^cosQi)] (2.3)

where the lag h is expressed in radians and X is the range.

A hole effect amplitude of 0.50 times the sill was found to be realistic after reviewing

examples in the literature. For the smoothly transitory covariance structure, a gaussian

model was chosen because it has parabolic continuity near the origin like the cardinal

cosine function. The model gaussian variogram and cardinal cosine variograms have

similar ranges (X) and identical sills as well as similar behaviour near the origin

To prepare the fields, one-dimensional arrays with a standard normal distribution (i.e.,

E[x]=0, Var[x]=l) were generated by a random number generator. The covariance model

was imposed using a modified form of the GSLIB simulated annealing program sasim

(Deutsch and Journel, 1992). Each model had 50 layers. The hydraulic conductivity (K)

values for flow models were obtained by transforming the standard normal values to a

lognormal K distribution where Y=ln(K), E[Y]=4.0 and Var[Y]=1.0. Single ID

realizations of each kind of model as well as the corresponding model variograms are

presented in Figure 2.2.

A unidirectional flow field parallel to the layers was simulated with the USGS finite

difference code Modflow (McDonald and Harbaugh, 1988). The effective conductivity of

each realization was found equal to the arithmetic mean of the layers, a trivial result

predicted using the concept of equivalent parallel flow in horizontal beds (Freeze and

Cherry. 1979). There was no difference in the calculated effective conductivity between

realizations, no matter which vertical structure was imposed. This effective conductivity

in parallel flow is independent of the position of the individual layers, so the choice of

20

Figure 2.2: One-dimensional conductivity fields used for the layered experiment and their experimental variograms. All have an expected value of ln(K) of 4 and a variance of 1.0. The realization in A is uncorrelated. The realization in B has a simple gaussian structure. The realization in C has a hole effect.

21

vertical covariance structure is inconsequential in predicting effective flow properties in

truly layered cases.

The flux distribution along the outflow face was examined for the gaussian versus the

hole-effect models of layering. A variogram of model cell fluxes along the outflow face is

shown for a single realization of each. The variogram values (normalized by the variance)

are shown in Figure 2.3. Not surprisingly, there is an imprint of the layer structure in the

flux field. The hole effect is pronounced in the variogram of the cell fluxes. It is

interesting to note that the variogram of the cell fluxes from the gaussian field also has a

hole-effect. This oscillation is presumed to be an artifact caused by a finite flow field only

~6 times the range of the underlying conductivity field. This result shows that while

effective flow properties are insensitive to layer structure, the distribution of fluxes are

affected. This spatial correlation of fluxes will control the pattern of contaminant

transport in space and time.

Experiment 2: Isotropic 2D Models

Three sets of 200 two-dimensional, unconditional fields incorporating an isotropic

random, gaussian, and hole-effect covariance model were produced by simulated

annealing. The same approximate range (X) of about 8 units for the gaussian and

hole-effect models were used. Each field was 50x50 blocks (6X x 6X) for a discretization

of 8 grid blocks per range. The conductivity values in each field are lognormally

distributed with E[Y]=4.0 and Var[Y]=1.0, where Y=ln[K].

The model variogram for the damped hole-effect realizations is the simple cardinal sine

model (Lantuejoul, 1994):

y(h) = 1 - sin(h)/h. (2.4)

22

Figure 2.3: Experimental normalized semivariograms for cell fluxes at exit faces of layered systems with a vertical hole-effect (solid) and gaussian (dotted) structure imposed. The hole effect of the K field manifests itself in the structure of cell fluxes.

23

where h is expressed in radians. The range is not defined for such a hole-effect model.

The cardinal sine model can be modeled in two dimension but has a maximum

amplitude of 0.212 times the sill. The cardinal cosine model can be made to have larger

amplitudes but the hole effect will only be present in one direction (Hohn, 1988). The

gaussian structure used matched the parabolic behaviour of the damped hole effect model

neair the origin. The range is comparable to the half-wavelength of the hole-effect model

and the sills are identical. Examples of the spatial distribution of Y with a gaussian and a

hole-effect variogram are shown in Figure 2.4.

The effective conductivity, Keff, of the 200 realizations of random, gaussian, and

hole-effect models were calculated using Modflow. ¥^s is calculated by imposing no-flow

conditions across two opposing sides of the fields and imposing a gradient across the

remaining two sides. When the model converges, the total calculated flux out the outflow

face is divided by the gradient to determine Keff.

The results are shown in Figure 2.5. The lowest mean value of Keff is for the random

structure. That the lowest 2D effective value is the randomly structured grid is predicted

by the results of Desbarats and Dimitrikopolus (1990). They showed that on 2D grids,

Keff has a lower bound equal to the geometric mean of interblock K values. They also

showed that the lowest variance of Keff occurs when the grid dimensions are infinite with

respect to the characteristic length (range) of heterogeneity. As the range increases

relative to domain size, the effective conductivity approaches the arithmetic mean

interblock K and the variance in Keff increases.

The variances of Keff between the gaussian and hole-effect models were statistically

compared and found to be identical at a 95% level of confidence. The difference in the

mean values of Keff is small though still statistically significant at the 95% level of

24

..........,..__. -

• • :.

• • :

tltl^l^S

Figure 2.4A: An unconditional isotropic field with a gaussian variogram structure.

25

• • • * • ' • * • • • • • . - . • • . , . ' .

IP

Figure 2.4B: An unconditional isotropic field with a hole-effect variogram structure.

26

Experimental Variograms for:

One Hole Effect Field —•

One Gaussian Field _- —

10 15 Lag

20 25 30

Figure 2.4C: Experimental variograms for single realizations of unconditional fields possessing a gaussian structure (Figure 2.4A) and a hole effect structure (Figure 2.4B).

A 150

Mean=48.409 S.D.= 1.329

50 55 60 65

Effective Conductivty (m/s)

B 100

e

§• 50

IV.

Jill. Mean= 53.778 S.D. = 2.341

45 50 55 60 65


100

u 2. 50 u

I I

Mean=55.634 S.D. = 2.299

j .

45 50 55 60 65


Figure 2.5: Comparison of distributions of calculated effective conductivity over 200 realizations of A. uncorrelated random fields, B. correlated random fields with a gaussian variogram, C. correlated random fields with a hole-effect variogram. The lowest mean effective conductivity is associated with the uncorrelated random fields. The mean value of effective K of the hole effect fields is significantly different than the effective K of the gaussian fields at a 95% confidence level. The variances are not significantly different.

28

confidence. This experiment corroborates Fogg's (1989) observation that hole-effects

have little impact on effective flow properties.

To shed light on the issue of connectivity of high K values across the fields, a

percolation experiment was performed. Silliman and Wright (1988) demonstrated a

relationship between a parameter called the "extreme path value (epv)" and percolation

thresholds on flow grids. The epv is defined as the cumulative probability associated with

the highest value of K on a grid for which there exists a connected path between adjacent

faces (Figure 2.6) along which all values of K are equal or greater than that value. The

epv is related to the percolation thresholds (e.g. Stauffer, 1985) by pc=l-epv where pc is

the percolation threshold (Silliman and Wright, 1988). The value of K associated with the

epv is the extreme path conductivity, or Kepv- The better connected a field is , the lower its

percolation threshold and the higher the value K«pv. A program to measure epv on a 2D

grid of K values is in Appendix A. Unlike true percolation experiments which consider

reversing pathways, this program only allows percolation along forward and lateral

pathways. This limitation is acceptable for consideration of solute transport in flow fields.

The distributions of epv values for the 200 random, gaussian, and hole-effect fields are

in Figure 2.7. The lowest values of epv (highest percolation thresholds) are associated

with the random structured grids. For comparison, a theoretical value of percolation

threshold on infinite regular 2D grids (King, 1990) is shown.

The distribution of epv's for the 200 gaussian and hole-effect fields are also shown.

They are greater than the epv's on random grids because the introduction of spatial

correlation has the effect of lowering percolation thresholds and raising epvs (Silliman and

Wright, 1988). There was found to be no statistically significant difference in either the

mean epv or the variance between the two distributions. This result indicates there is little

compartmentalization attributable to the hole-effect structure on these 2D grids at this

scale.

HighK

LowK

extreme path

extreme path conductivity (Kepv)

Kepv

Conductivity K

Figure 2.6: Illustration of extreme path value concept to measure connectivity of grid blocks. The extreme path is the connected path from one side of the grid to the other which contains the K^. The K^ is the highest value of block K for which there is a connected path, along which all other grid blocks are greater. The epv is the cumulative frequency or cumulative probability associated with K^. The percolation threshold, pc, is related to epv as pc=T-epv. The higher the epv, the lower the percolation threshold is and the better connected the high values of grid-block K are.

30

200

C U 0*100 0) u fa

EPV corresponding to King's (1990) percolation threshold on a 2D finite grid (0.403).

100

o a § 50

U-

iL Mean=0.4379 S.D. = 0.0281

0.2 0.3 0.4 0.5 0.6 0.7

Extreme Path Value 0.8

B

Mean=0.4903 S.D. = 0.0855

0.2 0.3 0.4 0.5 0.6 0.7


100

c u 3 50 cr <u

IX,

0.2 ••lIlL-

0.3 0.4 0.5 0.6 0.7

Mean=0.4830 S.D. = 0.0745


Figure 2.7: Distributions of extreme path values (epv) from forward percolation experiment for 200 realizations of A. uncorrelated random fields, B. correlated random fields with a gaussian variogram, C. correlated random fields with a hole effect variogram. All fields have E[Y]=4.0 and Var[Y]=1.0 where Y=ln(K). The mean epv in A is slightly higher than the theoretical value, perhaps because only forward percolation is allowed. There is no statistical difference at a 95% level of confidence in either the means or variances of the epv distributions shown in B and C.

31

The statistical behaviour of predicted hydraulic heads at each node over the families of

all realizations incorporating either covariance structure was examined. Smith and Freeze

(1979) demonstrated how introduction of a correlation structure increased the variance in

predicted heads over that of a random grid. To perform this experiment, steady-state

hydraulic heads under a constant gradient and parallel no-flow boundaries were simulated

with Modflow as before. The mean and variance in head values at all nodes were

computed and mapped over the families of 200 realizations.

The mean values of head at each node did not show any significant difference between

types of structure. The variances, however, were strongly affected. The mapped

variances in Figure 2.8 show higher values in the centre of the domain for the gaussian

fields than for the hole-effect fields. This difference in variance indicates there is less

difference between the realizations incorporating the hole-effect than those without. This

is a significant result because it suggests that enforcement of the hole effect is reducing the

space of uncertainty explored by these simulations.

The effective transport qualities of the covariance structures were tested. In fifty fields

of each type, one thousand unretarded solute particles were tracked through the

steady-state flow fields calculated by Modflow using the particle-tracking code Modpath

(Pollock, 1989). The dispersivities were calculated from the distribution of travel times to

the outflow face by (e.g., Desbarats and Srivastava, 1991):

« = f [S ] 2 P-5)

where a = the calculated longitudinal dispersivity, x is the longitudinal distance travelled,

CTat is the standard deviation in particle arrival time at x, and mat is the mean particle arrival

time at x.

Flow

No Flow Boundary 32

No Flow Boundary 10 20 30

No Flow Boundary

40 50

Flow

10 No Flow Boundary

20 30 40 50

Figure 2.8: Map of variance in predicted grid block head normalized by head difference across the domain. The variance was calculated over 200 realizations of A) structured randm fields with a gaussian variogram and B) structured random fields with a hole effect structure. No-flow boundaries were imposed on top and bottom boundaries. The hole effect fields have a lower variance in predicted heads.

33

The values of a were calculated for a selection of values of x (Figure 2.9). There was

no consistent variation between the mean values of calculated a over the scales of

observation studied. However, the variance in calculated a shows consistent reduction at

larger scales of observation when a hole effect covariance structure is used versus a

gaussian structure. This reduction of variance is consistent with the reduction in variance

in predicted hydraulic head values discussed above. The effect is not observed at short

scales of observation, perhaps because all variances are damped by boundary effects. The

reduction in variance means the space of uncertainty is being reduced and the hole effect is

adding an additional constraint on the stochastic simulations.

Experiment 3: 2D Panels with Vertical Hole Effect Only

As noted by Johnson and Driess (1989), hole-effect covariance structures are more

often noted in the vertical direction than in the horizontal. To test the effect of honouring

such a covariance structure, the hydraulic effects of a combined vertical hole-effect model

with a horizontal spherical model were compared to a vertical and horizontal spherical

model. A 5:1 horizontal to vertical anistropy was chosen. Otherwise, a high degree of

anisotropy would require a very large model to measure any hydraulic effects different

than the layered case of Experiment 1. As well, Einsele (1991) noted that allocyclic

sediments, which can possess a hole-effect type covariance structure, tend to have limited

horizontal extent.

Combining different covariance models in orthogonal directions on a lattice is possible.

However, it is not a trivial task to define the analytical expression for the covariance

structure off the main axes of the lattice (Schwarzacher, 1980). Different covariance

models can be combined in geostatistical estimation to accomodate directional differences

in covariance structure by giving each end member structure a high degree of directional

anisotropy and then combining them in a linear combination (e.g. Journel and Froidevaux,

1982). In this experiment, the power of simulated annealing was employed to directly

34

14U ^ ^ ^ ™

? T

s 120 --*-> 'w T

1 -

4) CX w 100 - -3

T

g ] J

•5 80 ; -• * — >

- ^ N

Cfl c o

T - ——~Z~^~^ -a 60 L

s£\ •"* "

£ - r i i i y ^ L_ i

3 1

o 1

C3

u 40 1

1 1 A ~

/ I _L

^ X i r i

20 I m I L

' jf t< I -*-t

rt

200 400 600 800 1000 1200

Scale of Observation (m)

Figure 2.9: Mean and standard deviation (error bars) calculated longitudinal dispersivity for 50 realizations each of correlated random fields with a gaussian structure (solid) and a hole-effect structure (dashed). All fields have E[Y]=4.0 and Var[Y]=l .0 where Y= ln(K m/s). The mean value of dispersivity increases with scale of observation but there is no significant difference in the mean values between types of field. The variance in mean dispersivity, as exemplified by the length of the vertical error bars (standard deviation), is greater for the hole-effect realizations at small scales of observation but lesser at large scales of observation.

35

enforce possibly non-linear combinations of variogram models on orthogonal axes

coincident with grid axes. Covariance relationships on off-axis diagonals were not

enforced. A similar choice appears to have been made by Ouenes and Bhagavan (1994)

and Sen et al. (1995) in their 2D models.

Thirty unconditional realizations each of the combined hole effect-spherical structure

and the simpler spherical model were generated. Conductivity in each of the fields is

lognormally distributed with E[Y]=4.0, Var[Y]=4.0, where Y=ln[K]. These fields can be

considered quite heterogeneous. The field dimensions are 4Xh by 5X.V, where X is the range

of the structure. Examples of each type of field with their experimental horizontal and

vertical variograms are in Figure 2.10

The effective conductivity of each field was calculated with MODFLOW as before.

No statistically significant differences were noted between the mean or variances in Keff in

either the vertical or the horizontal directions. However, the distributions of Keff when the

hole effect is present in the vertical are more multimodal (Figure 2.11) than the pure

gaussian distributions generated. The significance of this multimodality is not clear since

there are only thirty realizations in the sample populations.

As in the previous experiment, unretarded solute particles were tracked through the

simulated steady-state flow field. The longitudinal dispersivity in the horizontal was

calculated by Equation 2.5. The horizontal length of the fields is 4 A,, so only the most

severe channeling effects are likely to be overcome at the longest scales of observation

(Smith and Freeze, 1979). Nevertheless, no statistically signficant difference between the

mean calculated value of longitudinal dispersivity was discovered (Figure 2.12A). There

is a change in the variance in a. The variances in hole effect values are initially higher then

become significantly lower with scale of observation.

o G —

"C >

00

2

1.8

1.6

1.4

36

2

1.8

1.6

1.4|-

1.2

1

0.8

0.6

0.4

0.2

0 0

• p " - • • — i i

A

Vertical

B

Vertical •

/ *\ " * # .

/ * \ /

Horizontal

10 15 20 25 Lag

30 35 40 45 50

Figure 2.10: Experimental horizontal and directional variograms for single realizations of two types of correlated random fields. In A) the field has a spherical variogram structure in the vertical and in the horizontal. The anisotropy is 5:1. In B) the field has a hole-

effect variogram in the vertical but a spherical variogram in the horizontal. The anisotropy is also 5:1. The vertical variogram from A is shown in dashed for comparison. The realization corresponding to A is in Figure 2.IOC. The realization corresponding to Bis in Figure 2.10D.

37

•

m$mmm&>

c

,,/-.i"H:^l-...'-:f?

D

Figure 2.10 (continued): Unconditional isotropic fields with C) a spherical variogram structure in the vertical as well as the horizontal

and D) with a hole-effect variogram structure in the vertical.

38

10

c a 3 a1 u i—

fa

Mean=5687.5m/s SD=950.3 m/s

JilllliL. 20O0 4000 6000 8000

Effective K,, (Spherical-only fields)

10

i r.r fa

I 5

Mean=1515.3 m/s SD=237.1m/s

iuJlllIu 1000 1500 2(

1500 2000

Effective K , (Spherical-only fields)

2500

10

a a ID

(D » -

Mean=3.83 SD=0.83

JbJUllL Anisotropy (K/KJ (Spherical-only fields)

10

c

B Mean=5338.9 m/s SD=1121.4m/s

2000 4000

Effective K„ 6000 8000

D (Hole-spherical fields)

10

$

3 5

fa

Mean=1689.1m/s SD=262.0 m/s

_JkJjU_ 1000 1500 2000 2500

Effective K, (Hole-spherical fields)

10

c 3. 5

fa

Mean=3.26 SD=0.94

Anisotropy (K/KJ (Hole-spherical fields)

Figure 2.11: Histograms of effective horizontal and vertical hydraulic conductivities for thirty realizations each of correlated random fields with horizontal and vertical spherical variograms (A,C) and with horizontal spherical and vertical hole effect variograms (B,D). All fields have E[Y]=4.0, Var[Y]=4.0, where Y=ln(K). All fields have horizontal to vertical anisotropy ratios of 5:1. The hydraulic conductivity anisotropy ratios (Kh/Kv) are shown in E and F. There is not a significant difference in the means and variances of effective conductivities. The distributions with the vertical hole effect appear multimodal, but the significance of this is not determined by only 30 realizations.

39

o

c '•3 *-> '5b §

3

as U

"35

a. Q

"T3

a 5b 5 -J -a

2000

1500

1000

500

A: Horizontal Transport E[Y]=4.0, Var[Y]=4.0 Field Dimension 200x50 m (4\ by 5X,)

200 400 600 800 1000 1200 1400 1600 1800


35(1

300

B: Vertical Transport E[Y]=4.0, Var[Y]=4.0 Field Dimension 200x5 (4X„by5M

0m T

.

250 "

200

150

100

50 I

T

T

J.

1

J.

0


Figure 2.12: Calculated dispersivity from particle tracking versus scale of observation over thirty realizations each of structured random fields with a horizontal Gaussian variogram and either a hole-effect variogram (dashed line) or a gaussian variogram (solid line) in the vertical. The fields have a 5:1 horizontal:vertical anisotropy. Calculated horizontal dispersivities are shown in A. Calculated vertical dispersivities are shown in B. Mean values are joined by the line. Variances at each scale of observation areshown by vertical error bars centered on the mean. In all cases the calculated dispersivity increases with scale of observation. The mean values of dispersivity do not change with different combinations of structure. However, the variance in calculated vertical dispersivity is reduced when a hole effect variogram is present in the vertical.

40

The same effect is noted for flow in the vertical, parallel to the hole effect covariance

structure, though the reduction in variance is apparent at all scales of observation (Figure

2.12B). The boundary effects may be diminished because the domain is slightly larger

relative to X in the vertical than in the horizontal.

Extreme path values were calculated for all the realizations. The distributions are

shown in Figure 2.13. The distribution associated with a vertical hole-effect structure

shows a skew to higher values of epv (lower percolation thresholds). This result can be

interpreted to mean there is higher probability that extreme values will be connected in the

horizontal when a hole effect is enforced in the vertical at this scale and anisotropy ratio.

Discussion

The most significant results of the three experiments can be summarized as follows:

1. Mean effective properties like Keff and oil over multiple realizations are insensitive to

the incorporation of the hole effect covariance structure.

2. The variance in predicted effective properties like ¥^s and cci is reduced by

incorporating a hole effect. The variance in predicted heads also decreases.

3. Horizontal extreme path values are skewed rightward in the 2D case where a

vertical hole effect is enforced in the vertical. This skew means percolation thresholds are

being reduced, suggesting higher probability of connectivity of high values in the

horizontal.

4. That the mean effective properties are unaffected by choice of covariance model

indicates the hole effect has at most a second-order effect on flow and transport

41

a

a-

Mean = 3321 m/s S.D.= 1405.3 m/s

8000 10000 12000

10 "

K„

B

Mean = 3769 m/s S.D.= 1984.2 m/s

8000 10000 12000

Figure 2.13: Histograms of extreme path conductivities (Kepv) for horizontal paths over thirty correlated random fields, each with a spherical variogram in the horizontal and !5:1 horizontakvertical anisotropy. One type has a spherical vertical variogram (A) and the other type has a hole-effect vertical variogram (B) as shown in Figure 2.10. The appearance of a positive skew in horizontal Kepv when a hole-effect variogram is present in the vertical suggests higher probability of connectivity of extreme values and consequently better horizontal connectivity.

42

behaviours. Indeed, as the size of the flow domain increases relative to X, the choice of

covariance structure will become less important.

The decrease in variance in measured effective properties is significant. As noted by

Journel and Froidevaux (1982), incorporating a hole effect covariance structure, where

warranted, will bring in extra information on the spatial heterogeneity of a system. By

enforcing the hole effect as well as the other information embedded in a variogram (mean,

variance, range), the group of realizations of a stochastic simulator will be more

constrained. This will have the same effect as adding conditioning data - the variance

between realizations becomes reduced because more information is being honoured.

On the other hand, Experiment 3's skewed distribution of calculated epv's shows

better horizontal connectivity of high values. This result is the most significant of all the

results because transport of solutes and immiscible liquids is controlled by the connectivity

of high and low values. It shows that the transport qualities of 2D fields will be influenced

by the choice of the vertical hole effect model. This result means that geologic knowledge

about the nature of vertical bedding in an aquifer or petroleum reservoir affects

predictions of flow in the horizontal.

How can geologic study be used to help select a covariance structure? Listed below

are some guidelines for selection distilled from examination of the literature:

1. Purely random processes will produce no covariance at all measurable lags. No

covariance is represented by a pure nugget-effect variogram. Possible geologic candidates

include poured-in deposits, mass flows, debris flows (Schwarzacher, 1975). Stagnation

moraines and supraglacial deposits in glaciated environments may also be represented by

no measurable covariance structure.

43

2. Episodic processes that produce non-cyclic bedding successions should be modelled by

a simple covariance structure. MacMillan and Gutjahr (1986) demonstrate a correlation

between average bed thickness and ranges of vertical variograms in boreholes, for

example.

3. Autocyclic processes can be characterized by damped oscillatory co variance structures.

Geostatistical constructs of vertical sequences of beds from these types of environments

should be carefully scrutizined for real hole effects. Fluvial and deltaic deposits are

especially strong candidates because they incorporate feedback loops in their depositional

dynamics. Pseudo-cyclity may also be present in beds representing longer time scales

affected by eustatic fluctuations or tectonic pulses.

4. Truly cyclic or periodic sequences exist, especially in sedimentary records of beds

tied to diurnal, tidaL seasonal controls or to glaciation or Milankovitch-band climatic

variation. Glacial outwash deposits may also have a truly cyclic signature.

5. Non-equilibrium or turbulent systems may produce a fractal signature rather than a

standard co variance-type model (e.g., Painter, 1996). These models, not discussed here,

should also be considered when examining vertical bedding sequences.

44 References to Chapter 2

Aasum, Y., and M.G. Kelkar, 1991. An application of geostatistics and fractal geometry for reservoir characterization. SPE Formation Evaluation, March 1991, p. 11-19.

Agterberg, F.P., and I. Banerjee, 1969. Stochastic model for the deposition of varves in

glacial Lake Barlow-Ojibway, Ontario, Canada. Canadian Journal of Earth Science, vol.

6, p. 625-652.

Armstrong, M., 1982. Common problems seen in variograms. Mathematical Geology,

vol. 16, no. 3, p. 305-313.

Ashley, G.M., and T.D. Hamilton, 1993. Fluvial response to late Quaternay climatic

fluctuations, central Kobuk Valley, northwestern Alaska. Journal of Sedimentary

Petrology, vol. 63, no. 5, p. 814-827.

Bayer, U., 1985. Pattern Recognition Problems in Geology and Paleontology.

Springer-Verlag Lecture Notes in Earth Sciences 2, 229 pp.

Box, G.E.P, and G.M. Jenkins, 1976. Time Series Analysis - Forecasting and Control.

Holden-Day, San Francisco, 575 pp.

Desbarats, A.J., and S. Bachu, 1994. Geostatistical analysis of aquifer heterogeneity from

the core scale to the basin scale: a case study. Water Resources Research, vol. 30, no. 3,

p. 673-684.

Desbarats, A.J., and R. Dimitrikopolous, 1990. Geostatistical modeling of transmissibility

for 2D reservoir studies. SPE Formation Evaluation, p. 437-443.

45

Desbarats, A.J., and R.M. Srivistava, 1991. Geo statistical characterization of

groundwater flow parameters in a simulated aquifer. Water Resources Research vol. 27,

no. 5, p. 687-698.

Deutsch C.V. and A.G. Journel, 1992. GSLIB Geostatistical Software Library and User's

Guide:. Oxford University Press, New York. 340 pp.

Einsele, G., W.Ricken, and A. Seilacher, 1991. Cycles and events in stratigraphy - basic

terms: and concepts. In: Einsele, G., W. Ricken, and ASielacher, eds. Cycles and Events

in Stratigraphy. Springer-Verlag, p. 1 -22.

Fischer, A.G., and D.J. Bottjer, 1991. Orbital forcing and sedimentary sequences. Journal

of Sedimentary Petrology, vol. 61, no. 7, p. 1063-1069.

Fischer, A.G., and L.T. Roberts, 1991. Cyclicity in the Green River Formation (lacustrine

Eocene) of Wyoming. Journal of Sedimentary Petrology, vol. 61, no. 7, p. 1146-1154.

Fogg, G.E., 1989. Stochastic Analysis of Aquifer Interconnectedness: Wilcox Group,

Trawick Area, East Texas. Texas Bureau of Economic Geology Report of Investigations

No. 189, Austin, 68 pp.

Freeze, R.A., and J. Cherry, 1979. Groundwater. Prentice Hall, 604 pp.

Gelhar, L., 1986. Stochastic subsurface hydrology from theory to applications. Water

Resources Research, vol. 22, no. 9, p. 135S-145S.

Grant, C.W., D.J. Goggin, and P.M.Harris, 1994. Outcrop analog for cyclic-shelf

reservoirs, San Andres Formation of Permain Basin: stratigraphic framework,

46

permeability distribution, geostatistics, and fluid-flow modelling. American Association of

Petroleum Geologists Bulletin, vol. 78, no. 1, p. 23-54.

Harbaugh, J.W., and G. Bonham-Carter, 1970. Computer Simulation in Geology. John

Wiley & Sons, N.Y., p. 125.

Hein, F.J., and R.G. Walker, 1977. Bar evolution and development of stratification in the

gravelly, braided Kicking Horse River, British Columbia. Canadian Journal of Earth

Science, p. 562-570.

Holm, M.E., 1988. Geostatistics and Petroleum Geology. Van Nostrand Reinhold, New

York. 264 pp.

Isaaks, E.H., and R.M. Srivistava, 1989. Applied Geostatistics. Oxford University Press,

New York, 561 pp.

Jensen, J.L., P.W.M. Corbett, G.E. Pickup, and P.S. Ringrose, 1996. Permeability

semivariograms, geological structure, and flow performance. Mathematical Geology, vol.

28, no. 4, p. 419-435.

Johnson, N.M., and S.J.Driess, 1989. Hydrostratigraphic interpretation using indicator

geostatistics. Water Resources Research vol. 25, no. 12, p. 2501-2510.

JourneL A.G. and R. Froidevaux, 1982. Anisotropic hole-effect modelling. Mathematical

Geology vol. 14, no. 3, p. 217-239.

JourneL A. G., and Ch.J. Huijbregts, 1978. Mining Geostatistics. Academic Press, 600

pp.

47

King, P.R., 1990. The connectivity and conductivity of overlapping sand bodies. In: A.T.

Buller, E. Berg, O. Hjelmeland, J. Kleppe, O. Torsaeter, and J.O. Aasen, eds., North Sea

Oil and Gas Reservoirs-II. Graham & Trotman, p.354-362.

Kittridge, M.G., L.W. Lake, F.J. Lucia, and G.E. Fogg, 1990. Outcrop/subsurface

comparisons of heterogeneity in the San Andreas Formation. SPE Formation Evaluation,

September 1990, p. 233-240.

Lantuejoul, C , 1994. Nonconditional simulation of stationary isotropic multigaussian

random functions. In: M. Armstrong and P.A. Dowd, eds., Geostatistical Simulations.

Kluwer Academic Publishers, Boston, p. 147-177.

MacMillan, J.R., and A.L. Gutjahr, 1986. Geological controls on spatial variability for

one-dimensional arrays of porosity and permeability normal to layering. In: Lake, L. W.,

and H.B. Carroll, Jr., eds., Reservoir Characterization. Academic Press, p. 265-292.

McDonald, M.G., and AW. Harbaugh, 1988. MODFLOW: A Modular

Three-Dimensional Finite Difference Ground-Water Flow Model. Techniques of

Water-Resources Investigations of the United States Geological Survey, U.S. Geological

Survey. U.S. Department of the Interior.

MialL A.D., 1980. Cyclicity and the facies model concept in fluvial deposits. Bulletin of

Canadian Petroleum Geology, vol. 28, no.l, p.59-80.

Neton. M.J., J. Dorsch, CD. Olson, and S.C. Young, 1994. Architecture and directional

scales of heterogeneity in alluvial-fan aquifers. Journal of Sedimentary Research, vol.

B64. no. 2, p. 245-257.

48

Oertel, G., and E.K. Walton, 1967. Lessons from a feasibility study for computer models

of coal-bearing deltas. Sedimentology, vol. 9, p. 157-168.

Ouenes, A., and S. Bhagavan, 1994. Application of simulated annealing and other global

optimization methods to reservoir description: myths and realities. Society of Petroleum

Engineers Paper 28415.

Painter, S., 1996. Evidence for non-Gaussian scaling behaviour in heterogeneous

sedimentary formations. Water Resources Research, vol. 32, no. 5, p. 1183-1195.



frnite-difference ground-water model. U.S.G.S. Open File Report 89-381, 81 pp.

Power, B.A., and R.G. Walker, 1996. Alio stratigraphy of the Upper Cretaceous Lea

Park-Belly River transition in central Alberta, Canada. Bulletin of Canadian Petroleum

Geology, vol. 44, no. 1, p. 14-38.



Schwarzacher, W., 1980. Models for study of stratigraphic correlation. Mathematical

Geology, vol. 12, no. 3, p. 213-234.

Schwarzacher, W., 1993. Cvclostratigraphv and the Milankovitch Cycle. Developments

in Sedimentology 52, Elsevier, 225 pp.

Sen, M.K., A. Datta-Gupta, P.L. Stoffa, L.W. Lake, and G.A. Pope, 1995. Stochastic

reservoir modeling using simulated annealing and genetic algorithm. SPE Formation

Evaluation, vol.10, no. 1, p. 49-55.

49

Sharp, W.E., 1982a. Stochastic simulation of semivariograms. Mathematical geology, vol.

14, no. 5, p. 445-456.

Sharp, W.E., 1982b. Estimiation of semivariograms by the maximum entropy method.

Mathematical geology, vol. 14, no. 5, p. 457-474.

Shanley, K.W., and P.J. McCabe, 1994. Perspectives on the sequence stratigraphy of

continental strata. American Association of Petroleum Geologist Bulletin, vol. 78, no. 4,

p. 544-568.

Sillirnan and Wright, 1988. Stochastic analysis of paths of high hydraulic conductivity in

porous media. Water Resources Research, vol. 24, no. 11, p. 1901-1910.

Smith, L., and R.A. Freeze, 1979. Stochastic analysis of steady state groundwater flow in

a bounded domain 2, Two-dimensional simulations. Water Resources Research, vol. 15,

p. 1543-1559.

Srrvistava, R.M., and H.M. Parker, 1989. Robust measures of spatial continuity. In: M.

Armstrong, ed., Geostatistics ,Vol. I. Kluwer Academic Publishers, p. 295-308.

Stauffer, D., 1985. Introduction to Percolation Theory. Taylor and Francis, London., 124

pp.

Vail, P.R., F. Audemard, S.A. Bowman, P.N. Eisner, and C. Perez-Cruz, 1991. The

stratigraphic signatures of tectonics, eustacy and sedimentology - an overview. In: Einsele,

G., W. Ricken, and A.Sielacher, eds. Cycles and Events in Stratigraphy. Springer-Verlag,

p. 617-659.

50

Wilgus, C, S. Hastings, C. Ross, H. Posamentier, J. Van Wagoner, and C.G. Kendall,

1988. Sea-Level Changes: an Integrated Approach. Society of Economic Mineralogists

and Paleontologists Special Publication No. 42.

Zeller, E.J., 1964. Cycles and Psychology. In: D.F. Merriam, ed. Symposium on Cyclic

Sedimentatioa Geological Survey of Kansas Bulletin 169, Volume II. p. 631-636.

51

Chapter 3

Capturing Conceptual Models of Stratal Architecture in Synthetic Aquifers with Markov Chains and

Simulated Annealing.

Numerical solutions of flow and transport equations require estimates of hydraulic

parameters everywhere in the model domain. Assignment of grid-block parameters in

heterogeneous systems (geosystem modeling) can be done by zonation based on geologic

maps, by interpolation from measured data points, or by some sort of stochastic simulation

technique (e.g., Koltermann and Gorelick, 1996). Of these, geological maps often present

the only integrated model of heterogeneity at early stages of aquifer characterization when

hard information is limited. But since they are unique products of a geological synthesis,

they allow for little exploration of their associated uncertainty. Only when "hard" data

like core analyses, well tests, geophysical surveys, or outcrop-analogue studies become

available can most stochastic simulation techniques be used to assess flow and transport

uncertainty.

Markov fields may assist the geosystem modeler in some cases where hard data are

sparse. They can be constructed from geological models of stratal architecture based on

regional geology coupled with routine borehole or outcrop information. In this chapter, it

is demonstrated how to construct multi-dimensional Markov fields with geologically

meaningful structures using simulated annealing. Three Markovian expressions of stratal

architecture are demonstrated: hierarchical long-term stratigraphic memory (dependency),

C3'clicity, and directionality. Performance issues related to simulated annealing are

discussed in Chapter 6.

52

Building Markov Fields by Simulated Annealing

Markov Chains and Fields

A sequence of events wherein the present state of the sequence is contingent on the

state of the sequence at some time prior to the present is said to possess the Markov

property. If the stochastic process that produces the sequence is stationary, then we may

refer to a one-dimensional sequence of discrete events with the Markov property as a

Markov chain. The structure of a Markov chain can be summarized in a "transition

frequency matrix", wherein the frequencies of transition from any one state to itself or the

other states are tabulated (in columns) by state (the rows). If the frequencies are

normalized by the row totals, the matrix provides probability of transition from any state

to any other in a unit step. The sums along rows after normalization must equal one if all

possible contingencies are accounted for. Markov fields can exist where the Markov

property is present within a plane or volume (Lin and Harbaugh, 1984)

An example of a Markov transition probability matrix is shown in Table 3.1. This

matrix denotes the transition probabilities between three rock types - sandstone, shale,

limestone, when going vertically upwards within the Chester Formation of Indiana (from

Krumbein, 1967).

To: Sandstone Shale Limestone From: Sandstone 0.74 0.23 0.03

Shale 0.10 0.61 0.29 Limestone 0.05 0.38 0.57

Table 3.1 Markov transition matrix for an upward succession of strata.

53

The magnitude of the unit step or lag depends upon how the sequence is divided for

measurement. The choice is a nontrivial decision in Markov analysis of geologic

sequences for if time is being equated to vertical thicknesses, then allowances must be

made for varying rates of deposition within states (Schwarzacher, 1975). If the system's

observed state is contingent upon the state at a single location prior, we say the system has

a single dependency. If the observed state is contingent upon the system's state at two

times or locations prior, we say the system has a double dependency. Vertical sequences

of rock beds have been observed to possess statistically significant single or double

dependent Markov properties (Harbaugh and Bonham-Carter, 1970).

Analysis of the Markov properties of vertical sequences of rock strata was popular

with geologists in the 1960s and 1970s. Markov analysis was used to identify non-random

associations of rock facies in sedimentary sequences and transition matrices were used to

summarize facies associations in real deposits (Walker, 1979). This application fell into

disfavour as sedimentologists recognized the nonstationarity of depositional processes at

relatively short spatial scales in many types of sedimentary deposits (e.g., Miall, 1988).

Nevertheless, Markov analysis proved effective in the study of subtle stratigraphic

relationships such as double dependency (also called long-term stratigraphic memory),

depositional directionality (evidence of time's arrow in a process), and depositional

cyclicity.

It is these three latter properties of real rock sequences that invite further investigation

by hydrogeologists interested in characterizing heterogeneities of aquifer systems. It is

well known that simple two-point indicator or covariance-type models cannot completely

capture all the heterogeneity present in real geologic systems (e.g. Deutsch, 1992). On the

other hand, Markov transition matrices can encapsulate information about length scales,

relative frequencies of states, the likelihood of juxtaposition of different states, as well as

the three aforementioned properties.

54

Building Markov Fields with Simulated Annealing

One-dimensional simulation of Markovian sequences is straightforward using the

transition probability matrix and a random number simulator (Harbaugh and

Bonham-Carter, 1970). Krumbein (1967) adapted a one-dimensional Markov simulator to

the generation of a hypothetical two-dimensional transgressive-regressive shoreline

sequence. Lin and Harbaugh (1984) demonstrated the existence of two and

three-dimensional Markov fields and demonstrated a method to generate such fields.

However, their fields could not be conditioned to honour real observations. Moss (1990)

combined vertical Markov-chain simulation with prior estimates of horizontal

length-scales of states to simulate a North Sea oilfield reservoir. Lateral overlaps of

different states occupying the same cells in the model were eliminated using an erosional

hierarchy. Murray (1994) used simulated annealing to post-process indicator simulations

of a turbidite sequence to match observed vertical transition probabilities between states.

Similarly, Goovaerts (1996) improved reproduction of indicator cross-variograms, which

in a sense describe transition probabilities, in categorical fields by post-processing with

annealing. Carle and Fogg (1996) also used Markov transition matrices to model

indicator-type covariance and cross-covariance structures. Luo (1996) used a sequential

technique to generate Markov fields. While each of these prior efforts demonstrated

methods to generate Markov fields in geology, none investigated the relationships between

stratal architecture embedded in Markov transition matrices and hydraulic behaviour.

Following the work of Farmer (1992), workers such as Deutsch (1992), Ouenes and

Bhagavan (1994) and Datta Gupta et al. (1994) prepared stochastic fields by simulated

annealing. Simulated annealing is a global optimization method whereby an image or field

that honours an idealized set of control statistics is created from a random field or

structured field generated from some other method. The basic algorithm used in this study

is summarized in Figure 3.1.

55

Simulated Annealing (Metropolis algorithm)

1. Describe an ideal field. 4. Calculate initial value of 0 normalize to 1.000.

2. Generate trial field to be annealed. 5. Perturb by replacement or

swap of elements. 3. Define objective function, 0, e.g., the squared difference in control 6. Update 0

measures between images.

7. Accept perturbations that lower 0 ( Onew-OoW < 0).

8. Accept perturbations that increase O with p.d.f. P(accept) = exp(0old -Onew /T)

9. After a threshold number of acceptances, lower T and repeat.

10. Continue until either: - O reaches a

convergence criterion - O cannot be lowered.

Figure 3.1: Diagramatic explanation of how simulated annealing can create an image.

56

An idealized field is first characterized by some weighted combination of statistical or

other descriptive measurements. A trial field is then generated. The trial field can be a

totally random image matching the ideal global histogram or a structured field created by a

different algorithm. The same descriptors are calculated for the trial field. An objective

function, O, can be computed as the difference or squared difference between the ideal and

trial field descriptors. The components of the objective function may be weighted to assign

equal importance to small and large values or components with different units of

measurement (Deutsch, 1992). The trial field is then perturbed, usually by replacing the

value of one of the field elements with another drawn from the underlying global

histogram or by swapping two nodes at random. The objective function is recalculated.

Perturbations that reduce the value of the objective function are kept. Perturbations that

increase the value of the objective function are accepted with a probability that decreases

in proportion to the increase in objective function scaled by a parameter called the

"temperature". If the value of the objective function is normalized by the original value,

the cumulative probability density function governing acceptance of perturbations that

increase the value of the objective function is:

new \ r accept = expy Temperature ) C3-1)

If a predetermined number of perturbations are accepted (usually of the order of 10*N

where N is the number of elements in a field) before some maximum number of total

perturbations (of the order 100*N), the temperature is reduced by some factor less than 1

(e.g., 0.1). This procedure is repeated until the objective function falls below a threshold

or its value can no longer be reduced. For further details on the mechanics of simulated

annealing in stochastic field generation, the reader is referred to Jensen et al., (1997) and

Deutsch and Cockerham (1994) as well as the other references mentioned above.

57

A Markov transition matrix can easily be encoded as a control statistic in an annealing

objective function, making construction of multi-dimensional unconditional Markov fields

relatively straightforward. The form of the control statistic is that of the multipoint

histogram introduced by Farmer (1992).

For example, a single-dependency Markov transition matrix can be encoded in an

annealing objective function as a series of histograms. Two-point histograms denote the

expected number of transitions from any state i to any state j for a given lag separation in a

given direction (Figure 3.2). For the forward direction of a single-dependency Markov

chain, the expected number of transitions N, between states i and j in an nx*ny»nz field that

uses edge-wrapping to avoid edge effects (Deutsch and Cockerham, 1994) is:

N(i,j) = P(i).PG|i>nx.ny.nz (3.2)

where P(i) is the proportion of state i and P(j|i) is from the transition probability matrix. If

the Markov process is non-directional, then the same number of transitions N(ij) will

occur in the opposite direction. If the Markov process is directional, we can recalculate

the reverse transition probability matrix from the transpose of the forward transition

frequency matrix and get a different matrix. It is not necessary to enforce a histogram

structure in the reverse direction in a Markov field because this structure will be embedded

in the forward transition matrix.

More complex Markov structures can be built using multiple dependencies. For

example, in a double-dependent Markov chain the transition probability to a state k

depends upon the state j of the chain u-steps prior as well as the state i of the chain x-

steps prior to that. Harbaugh and Bonham-Carter (1970) suggested double- dependency in

bedding sequences may be the signature of two independent forcings on

58

I-" • • | , , . B

i« -> j i ras | , , . B

HP - • •

| , , . B

HP - • •

W!M$iflm& |

Trial field

J trial J ideal

1 Wk

1 eta Multipoint histograms for one direction

Fiigure 3.2: Illustration of multipoint histogram concept. The coloured grid represents the trial field. The direction is defined in terms of vectors on a cartesian grid. For each direction (and lag) being enforced, the number of transitions between all states are counted and stored in an array called the multipoint histogram. One such histogram exists for the trial field and one for the ideal field.

59

a depositional systems that act on different time scales. More complex structures

involving a hierarchy of dependencies greater than two can be imagined but are difficult to

justify in a geologic sense.

To translate a double-dependent Markov structure to multi-point histogram format

where [i-l and x is 1 or greater:

N(i,j,k) = P(i>PG|i)T*P(k[j|i)*nx.ny*nz (3.3)

where x= lag separating x(i) and x(j) in the series and u=l. P(i) and PG|i)T can be found by

simulation if only P(k|j|i) is known to start with. The reverse transition matrix can be

calculated by an appropriate transpose of the forward transition frequency matrices for

x=l only. For H,T>1, the reverse transition matrix and related conditional probabilities can

be estimated by one-dimensional simulation or simply not enforced in the annealing.

Deutsch and Cockerham (1994) discussed how conditioning data can be included in

annealed fields without introducing artificial discontinuities. Koltermann and Gorelick

(1996) commented that true Markov fields cannot be conditioned to match hard

measurements at more than one location or one side. Because simulated annealing is an

optimization method, it can accommodate a reasonable level of contradiction in the

objective function, allowing us to enforce a Markov structure in a categorical field while

attempting to honour observations at boreholes. By either varying the weight given to

honouring hard data or increasing the annealing convergence-tolerance on the Markov

structure, the modeller can increase or decrease the degree of replication of the desired

Markov structure in the geosystem model when conditioning data are available. One must

always remember that the Markov structure still only captures some essence of the real

heterogeneity of the system. Thus some deterioration in the annealed field's match with

the proposed Markov model can be

60

tolerated in the presence of conditioning data. If the Markov structure is unattainable

even within a geologist's tolerance for error, then the model structure may need review or

there is likely enough conditioning data present to try a different geostatistical approach.

To explore the effects of Markovian stratigraphic models on the permeability

architecture of simulated aquifers or reservoirs, two Fortran codes were written. The first

code, prephist, prepares multipoint histograms given a single or double-dependent Markov

chain with up to six categorical states for input into the annealing code. The annealing

code is called manneal. It uses either true simulated annealing or iterative improvement to

enforce Markov transition matrices along orthogonal axes in one to three dimensions. The

differences are discussed in detail in Chapter 6. The fields can be conditioned to replicate

hard measurements without discontinuities using the weighting scheme of Deutsch and

Cockerham (1994). The output of prephist is a parameter file called markov.par, which is

read by manneal. A separate parameter file, called annealpar, is used to control the

annealing parameters in manneal. The programs and sample input/output files are in

Appendix B.

Exploring Aspects of Markovian Stratigraphy

Markovian analysis of real rock sequences can reveal the existence of stratigraphic

relationships not well-captured by simple indicator or covariance-type structures, namely

double dependent contingency, stratigraphic directionality, and stratigraphic cyclicity. In

this section these Markovian aspects of stratal architecture are captured in

two-dimensional stochastic realizations. The conditions under which it may be

appropriate to extend Markov properties from one dimension, usually the vertical, to other

dimensions in a geosystem model are addressed in Chapter 4.

Double Dependency in Markov Fields

61

Recall that double dependency, also called second-order stratigraphic memory,

refers to a property of real strata whereby the state of the sequence at x significantly

depends upon the state of the sequence at two different prior locations, denoted x-\i and

X-J0.-T, in the sequence. A hypothetical, three-state double-dependent Markov transition

matrix with u=l and t=l , is:

P{k(x)|j(x-u)| Black(x-n-x)} k = Black Grey White

j = Black 0.850 0.075 0.075 Grey 0.075 0.850 0.075 White 0.075 0.075 0.850

P{k(x)|j(x-u)| Grey(x-n-x)} k = Black Grey White


P{k(x)|j(x-u)| White(x-u-x)} k = Black Grey White


Table 3.2: Hypothetical, double-dependent three-state Markov structure.

Embedded in this hypothetical, double-dependent structure is a single-dependent structure

(effectively a form of marginal probabilities):

P{j(x) i(x-u)} j = Black Grey White i = Black 0.800 0.100 0.100

Grey 0.210 0.500 0.280 White 0.230 0.250 0.520

Table 3.3: Single-dependent transition matrix embedded in Table 3.2.

62

A single realization of the double-dependent Markov structure was generated with

annealing. It is shown in Figure 3.3A. A single realization of the embedded

single-dependent Markov structure was also generated by annealing. The realization is

shown in Figure 3.3B. Visually, the main difference is there is less pixel noise in the

double-dependent field. Length-scales and proportions of each state are comparable in

each field.

Is this structure significant to the flow modeler? Two hundred unconditional

realizations (100x100 nodes) of both a single-dependent field and a double-dependent field

were generated by simulated annealing as above. The effective conductivities (Keff) of the

fields were calculated by using the USGS block-centered finite difference groundwater

flow model Modflow (McDonald and Harbaugh, 1989). For this experiment Kbiack=10

m/s, Kg[By=100 m/s, and KWhite=1000 m/s. The details of the Keff calculation are presented

in Chapter 2.

The distributions of Keff for each population are shown in Figure 3.4. Statistical tests

show the means and variances of the two population to differ significantly at a=0.06.

That the variance of Keff for the double-dependent fields is significantly less is likely due to

the added spatial constraint of the second Markov dependency. This effect is not unlike

the extra conditioning imparted by honouring a hole effect documented in Chapter 2. That

the mean Keff is also significantly less suggests there may be somewhat more flow

compartmentalization (less continuity of high values) in the double-dependent field at the

scale of this experiment. Whether this is an effect related to the aforementioned pixel

noise or is a real property of double-dependent Markov K-fields is not certain.

63

L , " J - a

Wr j 1"

W9mW£' *""J W*JI ft1 •' r p J ^*- « i ft1 •'

•M-H^V*' ••^jjSpS ^W%//M'- •"•fe^'^^I^L 1

1 ^ i d | J L x • .—i-Tr 1

l^dtBf/*!* f.T.IL rfk B

Figure 3.3: Two unconditional three-state Markov fields possessing the same single-dependency transition matrix. In A a double-dependency is also enforced.

64

30

Single-dependency

35 40 45 50 55

Mean Keff = 41.6246 m/s; Variance = 6.2612

30

Double-dependency

35 40 45 50 55

Mean Keff = 40.8042 m/s; Variance = 4.8077

Figure 3.4: Distribution of calculated effective conductivity over two hundred unconditional realizations of a three-state Markov field with the same single dependency but, in the second group, also having a double dependency in the transition matrix structure.

Directionality in Markov Fields

65

A strength of simulated annealing in the generation of stochastic fields is its ability to

reproduce directionality in spatial structures. Directionality cannot be generated in fields

built with symmetric variogram-based simulators (Deutsch, 1992). Directionality is

different from geostatistical anisotropy and nonstationarity. Geostatistical anisotropy

refers to a variation in range and/or sill values in a variogram structure as a function of

direction, though there is always a plane of symmetry implicit in the anisotropy.

Nonstationarity refers to a trend or change in the overall geostatistical properties with

translation in space. On the other hand, directionality is evidence of time's arrow in a

stationary depositional processes. Directionality implies the geostatistical structure varies

with reversal of direction, much as a movie is different run forward than backward. It is

present to a greater or lesser degree in most geologic materials.

Directionality in Markov probability models manifests itself as irreversibility in the

transition matrix. That is, the forward transition frequency matrix is not symmetric. Tests

for the significance of irreversibility in transition matrices exist (Powers and Easterling,

1982; Richman and Sharp, 1990). Because Markov chains can be irreversible, this aspect

of geologic heterogeneity can be easily captured in stochastic simulations with simulated

annealing.

Figure 3.5A shows an example of a stochastic field generated from an irreversible,

single-dependent Markov chain enforced in both the vertical and horizontal. In the

transition matrix (Table 3.4), we see a noisy directionality wherein, for example, state 2

(green) follows state 1 (blue) in the up direction with four times the probability of state 3

(red).

66

B

Figure 3.5: Unconditional fields with directionality enforced in A) both the vertical and the horizontal and B) in the vertical only. Cyclicity is also present.

67

P{j(x+l)i(x)} j=Blue Green Red i=BIue 0.75 0.20 0.05 Green 0.05 0.75 0.20 Red 0.20 0.05 0.75

Table 3.4: A Markov transition matrix with both directionality and cyclicity.

Directionality is evident in the realization as blue is most often succeeded in the

upward and leftward direction by green and most often preceded by red. Deposits which

have directionality that could be represented with directional Markov chains include

coarsening-upward, siliciclastic shoreline sequences, normally-graded turbidite sequences,

and fining-upward fluvial deposits.

Koltermann and Gorelick (1996) cite a comment by A.G. Journel that Markov fields

are difficult to use because it is not readily obvious what directionality means in the

horizontal. Indeed, in some cases it may be argued that vertical irreversiblity is not

relevant in the horizontal. Consider a case of stacked channel sequences. The upward

transition matrix may be directional in that facies sequences usually fine upwards. In the

horizontal however, there may be no similar directionality. A true Markov field can not

accommodate this discrepancy. With annealing however, we can generate fields that

posseses a hybrid Markovian structure such as a directional vertical structure but a

non-directional horizontal structure.

Figure 3.5B shows such a field. The same transition matrix as in Figure 3.5A was

enforced in the vertical. But for the horizontal, the off-diagonal elements of the transition

matrix were averaged together to form a perfectly symmetric transition matrix. This

procedure eliminates all directionality (Table 3.5). Thus, in the field shown in Figure

3.5B, there is a preferred upward sequence of blue-green-red while in the horizontal, there

is no directionality.

68

P{j(x+l)i(x)} j=Blue Green Red i=Blue 0.75 0.12 0.12 Green 0.12 0.75 0.12 Red 0.12 0.12 0.75

Table 3.5: The transition matrix of Table 3.4 with directionality removed.

Cyclicity in Markov Fields

Cyclic repetition of beds or sequences of beds in vertical directions is also common in

real sedimentary sequences. Geological cyclicity can develop in response to quasi-periodic

internal (autocyclic) or truly periodic external (allocyclic) forcings on the system. For

example, channel switching in fluvial systems is an autocyclic phenomenon. Seasonal

silt-clay rythmites in glaciolacustrine deposits is a classic example of allocyclic control on

sedimentation. Stacked sets of coarsening-upward siliciclastic parasequences show

cyclicity related to complex fluctuations of relative sea level at a multitude of time scales.

Cyclicity in single-dependent Markov Chains can be identified by the presence of negative

or complex eigenvalues (Schwarzacher, 1975). The matrix in Table 3.4 has two complex

eigenvalues, which confirms our visual impression of cyclicity evident in Figure 3.5 A. The

significance of cyclic stratal architecture in flow and transport simulation has already been

considered in detail in Chapter 2.

69


Carle, S.F, and G.E. Fogg, 1996. Transition probability-based indicator geostatistics.

Mathematical Geology, vol. 28, no. 4, p. 453-478.

Datta Gupta, A., L.W. Lake, and G.A.Pope. 1995. Characterizing heterogeneous

pemieable media with spatial statistics and tracer data using simulated annealing.


Deutsch, C.V., 1992. Annealing techniques applied to reservoir modeling and the

integration of geological and engineering (well test) data: Ph.D. thesis, Stanford

University, Calif.

Deutsch, C.V. and P.W. Cockerham, 1994. Practical considerations in the application of

simulated annealing to stochastic simulation. Mathematical Geology vol. 26, no.l, p.

67-82.

Farmer, C.L., 1992. Numerical Rocks. In: King, P.R., ed., Proceedings of the First

European Conference on The Mathematics of Oil Recovery, 1989. Oxford University

Press, p. 437-448.

Goovaerts, P., 1996. Stochastic simulation of categorical variables using a classification

algorithm and simulated annealing. Mathematical Geology, vol. 28, no. 7, p. 909-921.


Wiley&Sons, N.Y., p. 125.

70

Jensen, J.L., L.W. Lake, P.W.M. Corbett, and D.J. Goggin, 1997. Statistics for

Petroleum Engineers and Geoscientists. Prentice Hall, N.J., 390 pp.

Koltermann, C.E., and S.M. Gorelick, 1996. Heterogeneity in sedimentary deposits: A


Resources Research, vol. 32, no. 9, p. 2617-2658.

Kiumbein, W.C., 1967. Fortran IV Computer Programs for Markov Chain Experiments

in Geology. Kansas State Geological Survey Computer Contribution 13.

Lin, C. and J.W. Harbaugh, 1984. Graphic Display of Two- and Three-Dimensional

Markov Computer Models in Geology. Van Nostrand Reinhold, New York.

Luo, J., 1996. Transition probability approach to statistical analysis of spatial qualitative

variables in geology. In: Forster,A., and D.F. Merriam, eds. Geologic Modeling and

Mapping. Plenum Press, New York, p.281-297.

McDonald, M.G., and A.W. Harbaugh, 1988. MODFLOW: A modular three-dimensional

finite-difference ground-water flow model.

Miall, A.D., 1988. Reservoir heterogeneities in fluvial sandstones: lessons from outcrop

studies. Bulletin of the American Association of Petroleum Geologists, vol. 72, no. 6, p.

682-697.

Moss, B.P., 1990. Stochastic reservoir description: a methodology. In: Morton, A C , A.

Hurst, and M.A. Lovell, eds., Geological Applications of Wireline Logs. Geological

Society Special Publication No. 48, p. 57-76.

71

Murray, C.J., 1994. Identification and 3-D Modeling of Petrophysical Rock Types. In:

Yarns, J.M., and R.L. Chambers, eds., Stochastic Modelling and Geostatistics - Principles-

Methods, and Case Studies. AAPG Computer Applications in Geology, No. 3, p. 55-64.

Ouenes, A., and S.Bhagavan, 1994. Application of simulated annealing and other global



Powers, D., and R.G. Easterling, 1982. Improved methodology for using embedded

Markov chains to describe cyclical sediments. Journal of Sedimentary Petrology, vol. 52,

no. 3, p. 913-923.

Riehman, D., and W.E. Sharp, 1990. A method for determining the reversibility of a

Markov sequence. Mathematical Geology vol. 22, no.7, p. 749-761.

Schwarzacher, W., 1975. Sedimentation Models and Quantitative Stratigraphy. Elsevier,

New York, p. 259.


ed., Facies Models. 1st Edition. Geoscience Canada Reprints Series 1. p. 1-8.

72

Chapter 4

Informing Horizontal Markov Measures of Variability with the Vertical

Geosystem modeling is plagued by a lack of horizontal information pertaining to

heterogeneity. Horizontal categorical relationships, length scales, and connectivity of

extreme values of hydraulic conductivity cannot be defined from vertical boreholes unless

the heterogeneity exists at a scale greater than that resolved by vertical borehole spacing.

In some cases, descriptors of horizontal heterogeneity can be informed by horizontal wells,

geophysics, outcrop or subsurface analogues, or geological process simulators. Where

these are lacking, one option is to assume simple geometric anisotropy of heterogeneity

and apply vertical descriptors of heterogeneity to the horizontal after rescaling by a factor

usually in the order of 1:100 to 1:10000 (e.g., Painter, 1996).

Similarly, Doveton (1994) suggested that Markov measures of vertical variability from

boreholes may be transferred to the horizontal to inform a description of horizontal

variability. If Markov structures are to have a use in injecting geologic realism into

stochastic models, then this suggestion warrants further examination.

The geological justification for this transference comes through invocation of

Walther's Law of Facies Succession, an operating principle of stratigraphy. This idea has

also been explored in earlier works such as those of Krumbein (1967), Schwarzacher

(1975, 1980), and Lin and Harbaugh (1984). In this chapter, it is argued that if Walther's

Law of facies succession is used to justify transfer of Markov measures of vertical

variability in bedding sequences to the horizontal a coordinate transformation from

vertical space to vertical time must considered. This transformation is needed to honour

the principle of stratigraphic relationships encapsulated in Walther's Law. One beneficial

outcome of following this line of

73

reasoning will be the realization that the Markov fields constructed in a Waltherian

framework will conserve the sediment, time, and volume elements of a depositional

system. This effect goes some distance to satisfying Deutsch and Hewitt's (1996)

challenge to find ways to better honour geological concepts in stochastic modeling.

Coordinate Transforms in Markov Fields and Walther 's Law

Coordinate transformation in geosystem modeling is not new. Journel and

Gomez-Hernandez (1989) showed that forward stochastic simulation of bedding

heterogeneity within structurally deformed beds was better performed in a stratigraphic

rather than the original Cartesian coordinate system. A simple stratigraphic coordinate

system can be made by taking proportional distances between time surfaces to represent

vertical coordinates. This same approach has been taken in modeling heterogeneity in

offlapping clinoforms (e.g., MacDonald and Aasen, 1994) and complex channel sequences

(Deutsch and Wang, 1996).

The coordinate transformation necessary to transfer vertical Markov measures to the

horizontal in the context of Walther's Law is different from these rescaling methods.

Instead of modifying the entire space occupied by the categories in a Markov field, each

category is rescaled separately according to their relative depositional rates. To

understand the underlying need for this manner of rescaling it is necessary to consider

Walther's Law of Facies Succession.

According to Middleton's translation (1973), Walther's Law was originally stated as

follows:

'The various deposits of the same facies areas and similarly the sum of the rocks of

different facies areas are formed beside each other in space, though in cross-section

we see them lying on top of each other. As with biotopes, it is a a basic

74

statement of far-reaching significance that only those fades and fades areas can be

superimposed primarily which can be observed beside each other at the present

time."

Middleton emphasized that Walther's Law does not mean vertical sequences of facies

always reproduce the horizontal sequence, but rather that "... only those facies ... can be

superimposed .. which can now be seen developing side by side." Middleton continued: "

Walther's Law leads us to expect that each facies will show only certain transitions to

other facies, but it does not suggest that all of the genetically related facies can be

arranged in a single sequence, because some facies may represent alternatives at a given

stage in the development of any particular cycle".

This last statement gives the justification for using Walther's Law for translating

vertical Markov measures of variability to the horizontal. If the probability of any state

succeeding another in the vertical is equal to the probability that the states developed

adjacently at a given time, then the probabilities of any state being juxtaposed horizontally

should equal the probability that states are vertically superimposed, provided the

horizontal juxtaposition is coeval (italics mine). This argument frees us from the

deterministic and incorrect conclusion that Walther's Law implies that everywhere vertical

successions must be replicated in the horizontal. The probabilistic approach can also

accommodate minor erosional breaks in a succession as a component of random noise

(Doveton, 1994). This assumption will only work so long as the depositional process is

stationary at the scale of interest.

To ensure that horizontally adjacent facies are coeval, a multidimensional geosystem

model created from vertical Markov statistics under Walther's Law can be generated in a

time-space framework: vertical dimensions in time, horizontal coordinates in space.

75

Afterwards, vertical time coordinates can be transformed to spatial coordinates by

reseating vertical time thicknesses according to relative depositional rates and the effects

of compaction.

Mixed time-space coordinate systems are used in stratigraphy in a form called Wheeler

diagrams (after the work presented in Wheeler, 1958). Prior to vertical

back-transformation, Markov geosystem models could be regarded as synthetic Wheeler

diagrams. As mentioned above, Wheeler diagrams conserve sediment, time, and volume

in stratigraphic systems. Gaps in the sedimentary record due to erosion and nondeposition

are accounted for as a nondepositional or erosive state existing through time. It is

conceivable that if sufficient geochronological or biostratigraphic data exist to reconstruct

these states, they could enter a Markov model as a category in a mixed coordinate system

that is removed in the backtransformation.

A Demonstration

A fictitious first order Markov transition probability matrix involving three lithologic

states: sandstone, shale, and limestone, is shown in Figure 4.1 A. Assume the underlying

transition frequency matrix (Figure 4.IB) was derived from observation of vertical

variability in cores or outcrop. If the Markov chain was derived by making observations

of state at equal intervals of thickness, then the transition matrix will simply record a

geometric description of variability similar to the information stored multi-point histogram

(see Chapter 3).

Schwarzacher (1975) showed how a geometric Markov chain derived from equal

thickness measurements of bedding deposited at different rates may have no relationship

to the underlying stochastic process that deposited the beds. If constant rates of

76

sandstone shale limestone (cyan) (blue) (red)

sandstone 0.74 0.23 0.03 (cyan)

shale 0.10 0.61 0.29 (blue)

limestone \ 0.05 0.38 0.57 \ (red)

Original Transition Probability Matrix

sandstone shale limestone (cyan) (blue) (red)

173

: 45

16

54

273

121

B

7

130

182

Original Transition Frequency Matrix

C Rescaled Transition Frequency Matrix

sandstone shale limestone sandstone shale limestone (cyan) (blue) (red) (cyan) (blue) (red)

sandstone | 173 54 7 j 0.74 0.23 0.03 (cyan) 1 1

^~< x2 shale 45 ( 546 ) 130 j 0.06 0.76 0.18 (blue) I

x 3 ^ - ^ limestone ; i6 121 ( 576) 0.02 0.18 0.80

(red) V_y D

Transformed Transition Probability Matrix

Figure 4.1: Markov transition matrices for a hypothetical three-state system illustrating transformation to account for varying depositional rates. The relative vertical rates of deposition for the three states are assumed to be sandstone=1.0, shale= 0.5, limestone= 0.33. Colours correspond to those in Figures 4.2,4.3, and 4.4.

77

deposition are assumed for each category, then the vertical measurement interval can be

adjusted within each category to turn the observed sequence of observations from a space

series to a time series. A decompaction step may be needed as well to account for

reduction of original thickness due to consolidation (e.g., Bond and Komitz, 1984).

Let us assume for this demonstration that the relative rates of deposition of the three

lithologic states are 1:1/2:1/3 for sandstone:shale:limestone. The difference in relative

rates of deposition means that each unit vertical thickness of sandstone represents the

same depositional time as 1/2 unit of shale and 1/3 unit of limestone. The transition

frequency matrix can be rescaled according to reflect the state of the depositional process

over units of equal time instead of equal vertical space by multiplying the diagonal

elements by the inverse of their relative rates of deposition. The rescaled transition

frequency matrix is shown in Figure 4.1C. The temporally rescaled transition probability

matrix is in Figure 4. ID. In reality, lithologic units are unlikely to have been deposited at

constant rates so there will still be some distortion in the transition matrix (Schwarzacher,

1975).

According to the reasoning above, these vertical transition probabilities can be used to

directly inform horizontal relationships under Walther's Law provided that the system

possesses simple geometric anisotropy. Under this assumption, the transition probabilities

can be used directly with the method of synthetic field generation described in Chapter 3

to model geosystem heterogeneity. If simple geometric anisotropy is not realistic, then the

horizontal Markov transition probabilities would have to be rescaled to reflect the change

in length scales between vertical and horizontal. In the absence of horizontal observations,

such a rescaling would need to be informed by expert geological opinion, probably in a

Bayesian framework. An approach to the operational mechanics of such complex

rescaling is suggested in the work of Rosen and Gustafsen (1996).

78

A single 50x50 unit realization of a 2D field with the vertical and horizontal transition

probabilities in 4.1 A created by simulated annealing is in Figure 4.2. In this field, both the

vertical dimension and horizontal axes are in dimensions of space. A 5:1 horizontal to

vertical anisotropy is assumed. This field represents a single outcome of transferring the

geometric relationships observed in the vertical directly to the horizontal.

Compare this realization to the realization in Figure 4.3. In Figure 4.3, a realization

was made from the temporally-rescaled Markov transition matrix in Figure 4. ID. In this

case, the vertical axis is in units of time while the horizontal is in units of space.

Horizontal lines on this field represent isochrons. The field is back-transformed to vertical

space units in Figure 4.4. The formerly horizontal lines representing isochrons are now

complex surfaces. The top of the field in Figure 4.4, for example, was the horizontal line

at the top of the field in Figure 4.3.

The visual difference between the fields in Figures 4.2 and 4.4 is striking. The flow

and transport behaviours would undoubtedly be different between a family of realizations

prepared by each method.

No real data set is presently available to validate this approach.

79

Figure 4.2: A three-state, isotropic Markov field, vertical transition matrix as shown in Figure 4.1 A. Directionality has been removed in the horizontal..

80

<u

H .5 d .2

CO

c 6 Q 13 o '-£

Horizontal Dimension in Space

Figure 4.3: Three-state, isotropic Markov field, vertical transition matrix as shown in Figure 4. ID. Matrix was rescaled according to relative rates of deposition. Directionality has been removed from the horizontal.

81

Figure 4.4: The same field as in Figure 4.3, but vertical dimension has been backtrans-formed to spatial dimensions from dimensions of time through the relative rates of deposition. The vertical Markov structure is that in Figure 4.1 A. The horizontal Markov structure will be the same as the vertical only in transformed space or if the structure was measured parallel to each isochron (like the upper surface). For modeling the domain would be fully occupied by grid values.

82


Bond, G.R., and M.A. Komitz, 1984. Construction of tectonic subsidence curves for the

early Paleozoic miogeocline, southern Canadian Rocky Mountains: Implications for

sulbsidence mechanisms, age of breakup, and crustal thinning. Geological Society of

America Bulletin, vol. 95, p. 155-173.

Deutsch, C.V., and L. Wang, 1996. Hierarchical object-based stochastic modeling of

fluvial reservoirs. Mathematical Geology, vol. 28, no. 6, p. 857-880.

Doveton, J.H., 1994. Theory and applications of vertical variability measures from

Markov chain analysis. In: Yarns, J.M., and R.L. Chambers, eds. Stochastic Modeling

and Geostatistics - Principles, Methods, and Case Studies. American Association of

Petroleum Geologists Computer Applications in Geology, no.3. AAPG, Tulsa,

Oklahoma, p. 55-64.

Journel, A., and J. Gomez-Hernandez, 1989. Stochastic imaging of the Wilmington clastic

sequence. Society of Petroleum Engineers paper no. 19857.

Kirumbein, W.C., 1967. Fortran IV Computer Programs for Markov Chain Experiments


Lin, C. and Harbaugh, J.W., 1984. Graphic Display of Two- and Three-Dimensional

Markov Computer Models in Geology. Van No strand Reinhold, New York.

MacDonald, A.C., and J.O. Aasen, 1994. A prototype procedure for stochastic modeling

of facies tract distribution in shoreface reservoirs. In: Yarns, J.M., and R.L. Chambers,

eds. Stochastic Modeling and Geostatistics - Principles, Methods, and Case

83

Studies. American Association of Petroleum Geologists Computer Applications in

Geology, no.3. AAPG, Tulsa, Oklahoma, p. 91-108.

Middleton, G.V., 1973. Johannes Walther's Law of the Correlation of Facies. Geological

Society of America Bulletin, vol. 84, p. 979-988.








Geology, vol 12, no. 3, p. 213-234.

Wieeler, H.A., 1958. Time stratigraphy. AAPG Bulletin, Vol 42, no. 5, p. 1047-1063.

84

Chapter 5

Markov Characterization of Vertical Variability in a Complex Aquitard Unit: Gloucester Waste Disposal Site, Ontario.

History of the Site and General Geologic Setting

The Gloucester waste disposal site is a contaminated site located immediately south of

the Ottawa, Ontario, airport (Figure 5.1). The site is underlain by layered and

interfingering glacial sediments mantling limestone bedrock (Figure 5.2). Federal

agencies, hospitals, and universities disposed of various waste chemicals in an unlined

gravel pit at the site from 1962 until 1980 (Jackson et al., 1985). A contaminant plume

composed of various dissolved organic compounds (Figure 5.3) was identified in the

glacial sediments beginning in 1979 (Jackson et al., 1985). In the following decade the

site was the subject of numerous hydrogeo logic investigations by researchers, government

scientists, and consultants. The groundwater is presently under active remediation with a

combined managed gradient / pump-and-treat system. Issues of site remediation are

discussed by Jackson et al. (1991) and Gailey and Gorelick (1993).

The site geology is considered to be a layered system consisting of (from bedrock to

surface) limestone bedrock overlain by a discontinuous layer of basal till; a confined,

coarse sand-and-gravel aquifer termed the "Outwash Aquifer"; an interbedded silt, clay,

and fine sand aquitard termed the "Confining Layer"; and an unconfined, fine sand aquifer,

termed the "Surficial Aquifer" (Figure 5.2). Detailed stratigraphic subdivision of the

lateral interfingering and vertical interbedding of sands, silts, and clays in the Outwash

Aquifer and its transition to the overlying Confining Layer was not pursued beyond the

original geological investigations (e.g., French and Rust, 1981; Geologic Testing

Consultants, 1983) because of the complexity of these sediments (Jackson et al , 1985).

85

Ottawa International Airport

North

Gloucester Waste Disposal Site

Linear ridge of fluvioglacial deposits

o

kilometres

Figure 5.1: Schematic illustration of location of Gloucester waste disposal site relative to Ottawa airport and extent of linear sand ridge (after Jackson et al.,1985).

86

Surficial Aquifer Confining Unit

Outwash Aquifer

• i i i i

Basal Till Lenses Limestone Bedrock

Figure 5.2: Schematic illustration of stratigraphic model of the Gloucester waste disposal site (after Jackson et al., 1985). No scale.

87

West East

Former Disposal Trench

TTTTTTT^i i

Contaminant Plume Direction of Groundwater Motion in Outwash Aquifer — — — — — —

Figure 5.3: Schematic illustration of contaminant plume in the Outwash Aquifer at the Gloucester waste disposal site (after Jackson et al., 1985). No scale.

88

This chapter has three objectives. The first objective is to characterize the layered

heterogeneity of the Confining Layer with Markov statistics. The source data for the

Markov analyses comes from sediment cores collected on the site in conjunction with

Environment Canada in October, 1995. The second objective is to relate the Markov

statistics to the geologic model proposed by others. The third objective is to characterize

hydraulic properties of the sediments. The results will be used in Chapter 6 to test the

idea that Markov statistics can be an effective method for using geologic information to

constrain stochastic simulations of permeability fields.

General Geology of the Area of the Gloucester Waste Disposal Site

The Ottawa region is blanketed by glacial sediments of Late Wisconsin age. Most of

these sediments are tills, but glaciofluvial, glaciolacustrine, and glaciomarine sediments are

also abundant. Overlaying much of the glaciolacustrine/glaciomarine sediments are

deposits of the Champlain Sea, a marine incursion which followed Wisconsan deglaciation.

Eskers and subaqueous delta-fan complexes are common. These features mark the

drainage patterns and outflows of subglacial meltwater conduits (Fulton et al., 1987).

In the area south of Ottawa, Rust and Romanelli (1975) identified a series of elongate

raised ridges of sand and gravel. They argued on the basis of sedimentological evidence

that these landforms formed under water at an ice front. The height of these ridges

suggests the water could have been up to 100 m deep. They reported a consistent spatial

pattern of sediments associated with these ridges. The ridges themselves have a core of

sorted sand and gravel showing clear evidence of deposition by water currents. The sands

and gravels grade rapidly, both laterally and vertically, into stratified sands on the flanks of

the ridges. These beds were interpreted to be topset beds of coalescent subaqueous

outwash deltas. Rust and Romanelli (1975) presumed that further from the ridges, the

stratified sands would grade into fine

89

sands and silts of the delta aprons and then into silty clays of a deep proglacial lake.

Soft-sediment deformation, graded sequences, climbing ripples, and deeply incised

channels in outcrops of the stratified sand suggest seasonal flood events and mass wasting

on the tops and flanks of subaqueous deltas.

Rust (1977) reported on mass-wasting processes on these subaqueous outwash deltas

in the Ottawa area. He used sedimentological evidence to bolster the argument of Rust

and Romanelli (1975) that the nonfossiliferous interbedded glaciofluvial silts, sands and

gravels overlying bedrock in the Ottawa area represent deposition in subaqueous

esker-outwash fans below wave-base in a proglacial lake or the Champlain Sea. He

reasoned that much of the heterogeneity observed is due to a combination of deposition in

distributary channels on the outwash fans and mass wasting processes on their flanks.

More specific to the Gloucester Waste Disposal Site, French and Rust (1981) showed

the disposal trench to be located on the northwest terminus of one of these sand and

gravel ridges. They reported on the detailed stratigraphy in the immediate vicinity of the

disposal trench. The Confining Layer is not present at the trench, so their core

descriptions did not include any of the stratigraphic features of interest to this study.

Extensive drilling during subsequent investigations allowed more complete mapping of the

gross lithostratigraphic units under the site (Geologic Testing Consulting, 1983, revised by

Jackson et al., 1985).

The final conceptual model divided the site stratigraphy into five major units. The

bedrock is a dark shaley limestone. It is overlain by a discontinuous layer of dense, coarse

basal till. Atop the bedrock and basal till is a 25-m thick succession of complex

interstratified silts, sands, and very poorly sorted gravels. The axis of deposition of this

unit was along the northwest-southeast gravel ridge which skirts the south and west

corner of the site. The gravels do not extend far beyond the ridge, indicating rapid

90

deposition in slack water and/or mass flows. Sands and silts persist more distally from the

gravel ridge and are interpreted to be more distal facies of subaqueous outwash fans.

Graded sequences noted in some cores provide evidence of episodic deposition (Jackson

etal., 1985).

There is a transitional contact between the thick sand and gravel unit and the overlying

unit of clayey silts and silts. The transition is marked by a gradual decrease in the

proportion of coarse layers. Stratigraphic correlations (e.g. Jackson et al., 1985) show

lateral interfingering of this unit with the underlying unit, attesting to an episodic nature of

deposition. The transition zone marks the retreat of ice from the area and the deposition

of fine sediments in deep water. The clay and silt unit is not found on the west side of the

site.

The clay and silt unit is overlain by a frne-to-coarse sand unit which subcrops beneath

the soil cover over the entire site. Jackson et al. (1985) note the presence of marine fossils

in the sand within the Gloucester area. They considered this to be evidence that the unit

consists of marine sediments. The sands are related to shoaling of the Champlain Sea. An

unconformity separates the surficial sand from the underlying glaciofluvial-glaciolacustrine

complex (Rust and Romanelli, 1975).

The 1995 Sampling and Analysis Program

A sampling program to investigate the heterogeneity of the Confining Layer was

czirried out on an uncontaminated part of the site in October, 1995. Sediment cores were

collected with a 1-in ID split-spoon sampler with acrylic core sleeves run down a

hollow-stem auger. The drill crew and material were provided by Environment Canada.

Access to the site was granted to Environment Canada for sediment coring only. A

precondition for access was that no onsite hydraulic tests (wells, slug tests, etc.) or

onsite/offsite chemical analyses of the samples were to be done.

91

Cores were collected continuously from 1.5 m (5-ft) depth to the occurrence of sand

heave marking the breach of the Outwash Aquifer. Core recoveries varied between 0 to

100 percent, but were typically around 73%. The general area for drilling was selected on

the: basis of the geologic fence diagram presented by Jackson et al. (1985, their Figure 9) ,

the lack of contamination, and other site constraints. The area was previously mapped as

being near the southwest extent of the Confining Layer. It was hoped that the cores

would capture some of the lateral interfingering of the Outwash Aquifer and the Confining

Layer. Actual locations were constrained by drill-access considerations (uneven ground,

trees, etc.) Auger and sampler refusal due to large rocks was infrequent but several holes

had to be repositioned and redrilled. Relative borehole locations are shown in Figure 5.4.

Survey data for locations and elevations are found in Appendix C.

In the laboratory, the cores were subdivided into ~10-cm lengths for measurement of

vertical hydraulic conductivity (Kv) using a falling head permeameter. The cores were

measured and weighed when saturated, then extruded for detailed description under a

microscope. The cores were described at a 2-mm scale. This fine scale was necessary to

define all visible heterogeneity, including sand stringers in silty clays, that might affect

flow and transport. For Markov analysis, the core descriptions were converted into a

depth-categorical data set of six unique lithotypes (discussed below). The samples were

then photographed for archiving and then dried and reweighed for porosity determination.

Lithotypes

Six lithotypes can be visually recognized in the sediment cores from the Gloucester site

Confining Layer. These lithotypes are generally described in Jackson et al. (1985). The

lithotypes are:

92

100

50

a

C o Z

-50

•100

-150

10 X .» 11

»* * — » 8 ^ * 9 7

*--MW77-P1

14*

15 *

12

13

•100 -50 50 100 150

Easting (m)

Figirre 5.4 Map of relative locations of boreholes at Gloucester Waste Disposal Site. Cross-section X-Y is shown in Figure 5.5. Cross-section Y-Z is shown in Figure 5.6.

93

1. Medium to coarse grained, well-sorted, subangular quartz sand with very minor

amounts of biotite, amphiboles, feldspars and rock fragments. Bedding appears massive

with no structures visible in core. Medium to coarse sand from the Outwash Aquifer was

invariably totally disturbed because of sand heave during sampling. Basal contacts tend to

be sharp.

2. Fine grained, well-sorted quartz sand, often silty. Sometimes the fine sand occurs

with subrhythmic horizontal colour variations from light brown to dark brown The

colour variation is due to varying amounts of mafic minerals, suggesting minor episodic or

cyclic variations in sediment carrying-capacity of the depositional system. The fine sand

occurs on two scales: as massive or rhythmically coloured beds on the core scale and as

stiingers or thin interbeds within silty clay.

3. A medium brown to grey quartz silt with fine sand. The silt could be distinguished

from fine sand only under the microscope. One diagnostic character of the

water-saturated silt is its tendency to exhibit quick conditions when shaken.

4. Olive grey to dark grey, stiff silty clay. The silty clay is mostly massive but

sometimes mottled and rarely burrowed. Burrows are filled with silty fine sand. Fine sand

partings and stringers occur often. Single pebbles are occasionally found in silty clay and

clay lithotypes.

5. Olive grey to dark grey stiff clay. The degree of stiffness and visual lack of silt size

particles distinguishes the purer clay lithotype from the silty clay. Fine sand stringers

occur. Contacts between silty clay and clay lithotypes are gradational.

6. An unsorted fine to coarse sand, usually with pebbles and a high proportion of rock

fragments and mineral grains other than quartz, usually bound in a stiff silty clay matrix.

The mixture has sharp basal contacts and often grades upwards into a

94

medium-coarse sand. The lithotype is given the label "diamict", implying a glacial meltout

origin.

Photographs of representative parts of core showing these lithotypes are in Appendix

C. Detailed core logs are also found in Appendix C.

A west-east cross-section through the boreholes is shown in Figure 5.5. A

north-south cross-section is shown in Figure 5.6. The lack of easily correlated beds or

units between even close spaced boreholes attests to the high degree of heterogeneity in

this deposit. No single unit or marker bed could be identified which could help to

correlate horizontal beds within the deposit.

Markov Descriptions of Vertical Variability

A Markov process can be defined as "one in which the probability of the process being

in a given state at a particular time may be deduced from knowledge of the immediately

preceding state" (Harbaugh and Bonham-Carter, 1970, p. 98) In this way, a stochastic

process with the Markov property is said to possess a finite memory. A Markov chain is a

sequence of discrete states that are the outcome of a Markov process.

The Markov property of a sequence can be captured in a matrix of probabilities known

as the transition matrix. The transition matrix comprises an equal number of rows and

columns. In each row are tabulated the probabilities that each of the possible outcome

states will follow the row state at the step prior. Transition matrices are prepared by

tallying all observed transitions in a Markov chain for a chosen sample spacing or lag

(summarized in the tally or transition frequency matrix), then summing the rows and

dividing each element by their respective row totals. Because they represent exhaustive

probabilities, the rows of a transition matrix must each sum to

X West 8: 2=98.139 m

95

Y East

7:Z=97.95ml0:Z=97.97m

9: Z=97.78 m

ll:Z=97.58m

Interpreted Boundary of Confining Tayer.

Horizontal Scale (m)

Figure 5.5: Structural cross-section X-Y highlighting strata of the Confining Layer, Gloucester Disposal Site. See Figure 5.4 for section location.

96

North Y

South

15:Z=97.57m

14: Z=97.42 m 13: Z= 97.20 m

12: Z= 97.22 m

Horizontal Scale (metres)

6: Z= 97.31m

\ /

Datum Z=94m

Interpreted Boundary of Confining Layer

Medium to coarse sand Fine sand

I Silt Silty clay Clay

;— Diamict X j Missing Section

Figure 5.6: Structural cross-section Y-Z highlighting Confining Layer, Gloucester Disposal Site

97

one. Complex Markov chains that have significant transition probabilities between states

at more than one lag can be captured in a hierarchy of transition matrices.

Chi-squared tests are usually used to test the hypothesis that a Markov chain is not

different from an independent series of events. The usual test statistic is:

Z2 = I,.Z.<W ( 5 1 )

where Oij is the observed number of transitions and E„ is the expected number of

trcinsitions. To arrive at an estimate of Ey, it is usually necessary to have an estimate of

the independent probabilities of each state. The independent probabilities are estimated by

raising the transition matrix to a sufficiently high power so that the elements in each

column no longer change with further powering. The identical elements in each column

vector will equal the independent probability of any of the states occurring. The transpose

oi* any row vector in this matrix is the marginal probability vector. Other properties and

peculiarities of Markov transition matrices in geology are discussed extensively by

Harbaugh and Bonham-Carter (1970), Agterberg (1974), Schwarzacher (1975), Davis

(1986), and Doveton (1994). There is a huge volume of literature on applications of

Markov chains in statistics outside of geology. A helpful introduction to Markov chains

(built around MATLAB) is found in Kao (1996).

Rare events in geological sequences pose a special problem for Markov analysis. In

order to properly apply a chi-squared test to a Markov transition matrix, there should be

at least five events for each possible transition in the matrix (Davis, 1986). Yet it is

possible to have very rare events in a geological sequence that have significance out of

proportion to their number of occurrences. It is also possible simply to not have enough

outcrop or core to witness all possible transitions with a minimum frequency of five. In

these cases, the analyst is faced with five options. The

98

analyst can disregard the rare events entirely, blend them with another similar category,

blend them with all the other rare events, apply the chi-squared test with the data as is, or

don't apply it at all. None of these choices is really satisfying. For the purposes of this

work, I have chosen to deal with rare events by the second last option, accepting the fact

that the chi-squared approximation of these test statistics will deteriorate.

Applying a Markov chain model to describe vertical variability in geologic materials

implies the existence of a stochastic depositional process operating continuously or at

regular discrete intervals of time. However, geologic records normally include gaps

representing intervals of erosion or non-deposition, across which the depositional process

may either resume unchanged or reorganize completely. Miall (1985) called into question

the general applicability of Markov chain models to description of vertical variability

because of the violation of assumed stationarity of depositional process across significant

geologic breaks. Doveton (1994) also argued that the problem of stationarity across

geologic breaks can be severe if the breaks are truly significant, but he suggested that

minor breaks can be absorbed as noise in the transition matrix. Because of the hierarchical

nature of bounding surfaces within vertical sequences, the decision to proceed with a

Markov analysis must be as much a geologic as a statistical one.

Embedded Markov Chains

A special form of the Markov Chain is the embedded form. An embedded Markov

chain counts only transitions between different states. Same-state transitions are not

counted. An embedded transition matrix can be made from a standard first-order tally

matrix by setting the diagonal elements to zero before calculating the row totals and

transition probabilities. In an embedded Markov Chain, all length-scale information is

99

lost. The embedded chain is thus immune to any distortions due to sample interval or

varying sedimentation rates.

Embedded forms are used in geological analysis to isolate non-random associations of

categories or facies in ancient depositional systems (e.g. Walker, 1979). A geosystem

model generated by the techniques discussed in Chapter 3 and 4 of this dissertation will

have such relationships embedded within it. It should thus be incumbent upon the modeller

to investigate these relationships and demonstrate that they can be corroborated with

independent geologic reasoning or evidence. Otherwise the geosystem model may be in

violation of the natural system even though all of the control statistics are honoured.

Analysis of the embedded form may also identify hidden complexities of the natural model

that may be important in considering flow and transport behaviour outside of the

simulation exercise. This connection to geology is an advantage that Markov chains have

over empirical multi-point histograms or covariance structures more commonly used in

geostatistical simulation.

The Markov analysis of the Gloucester cores proceeds as follows. First, the data are

examined for homogeneity of process. Homogeneity is the spatial equivalent of the

stationarity in time series analysis. Then the conventional Markov structure is

investigated. This analysis did not reveal any structure beyond length scales related to

bedding. Filtering fine-scale noise and temporal rescaling did not affect this observation.

The embedded form offered more clues to an underlying Markov process in the

Gloucester Confining Layer but the signal-to-noise ratio is still too low to be conclusive.

An examination of the hydraulic properties of the lithotypes is then presented. The results

are related to the depositional model for the Gloucester site as a conclusion to this

chapter.

100

Homogeneity of Depositional Process

The grouping of all observations into one data set is based on the assumption that the

process of formation remains constant with translation in space. This invariance with

spatial translation is known as homogeneity. An invariance with temporal translation is

known as stationarity, though in geostatistics the term stationarity has been adopted for

both spatial and temporal invariance of process. It is often argued in the geostatistical

literature that stationarity or homogeneity is more properly viewed as a property of the

geosystem model, a simplified representation of nature, and not necessarily a property of

nature itself. Nevertheless, it remains good practice that an analyst search for geologic or

geostatistical evidence of homogeneity/stationarity or the contrary as part of a thorough

cliaracterization of a data set.

The area from which the data come is less than one hectare (100m * 100m). This

small area suggests that geologically there is strong reason to expect homogeneity of

process. To test the hypothesis, Powers and Easterling (1982) define a chi-squared test

for embedded Markov matrices. Their test statistic for homogeneity of process is

approximately chi-squared distributed and is of the form:

, 2 V V V (.nti~Etij) X =*t±i IjW Ej (5-2)

where

Etij = £tn+ij (5.3)

and E j is the expected number of transitions between i and j in matrix t, ritij is an element

of the t* m x m transition frequency matrix, and + indicates summation over all the

matrices. This complex factor is simply the average value of a given transition

101

taken over all the matrices being compared times the row sum for an individual matrix.

This test statistic has t(m-l)m degrees of freedom.

The Confining Unit data were divided into two subsets. The East subset included all

tramsitions observed in cores from boreholes 2, 12, 13, and 14. The West subset includes

all transitions in cores from boreholes 1,9, 10, and 11. Further subdivision is not

practicable given the small values of some of the elements. The embedded transition

submatrices are shown in Tables 5.1 and 5.2.

State (East Subset)

Medium Sand

Fine Sand

Silt Silty Clay

Clay Diamict

Med. Sand 0 1 2 1 0 0 Fine Sand 0 0 10 50 9 4

Silt 2 7 0 10 3 2 Silty Clay 0 55 11 0 2 5

Clay 0 5 3 7 0 0 Diamict 2 6 1 2 1 0

Table 5.1: Markov transition frequency matrix, east subset of Confining Layer cores

State (West Subset)

Medium Sand

Fine Sand

Silt Silty Clay

Clay Diamict


Silt 3 15 0 6 0 0 Silty Clay 1 35 11 0 5 5

Clay 1 7 0 6 0 0 Diamict 2 5 0 5 0 0

Table 5.2: Markov transition frequency matrix, west subset of Confining Layer cores

If we relax the restriction that only matrix cells with a value of 5 or greater should go

into calculation of the test statistic, then the test statistic for homogeneity of

102

process between these two submatrices is 31.10 for 24 degrees of freedom. The test

statistic is less than 37.65, the critical value of %2 for 24 degrees of freedom at a

confidence level of 95%. There is no statistical reason for rejection of the hypothesis of

homogeneity of process under the assumption that transition frequencies less than 5 are

significant and can be used directly.

The decision of vertical stationarity is a geologic one. The top of the Confining Layer

is believed to be an unconformity so any Markov analysis of vertical variability cannot

include sediments of the Surficial Aquifer. The transition between the Outwash Aquifer

and the Confining Layer is reportedly gradational so the selection of the base of the

Confining Layer is arbitrary. The transition from the medium sand of the Outwash

Aquifer to the interbedded silts and clays occurred rapidly in most boreholes in this study,

so selection of the base of the Confining Layer was defined as the first appearance of the

silty clay lithotype. Within the Confining Layer there are no vertical trends evident or

varved clays indicating a substantial deepening of water or change in the style of

deposition. Therefore the geologically reasonable assumption is made that the process

underlying the deposition of the sediments of the Confining Layer is stationary in time

between the top of the Outwash Aquifer to the unconformity at the top of the Confining

Layer.

Conventional Markov Description

Markov statistics of vertical variability can be collected by recording categorical

information on an equal vertical thickness or equal time-interval basis. If the information

is recorded on an equal vertical thickness basis, the resulting Markov transition matrix will

capture information on the length scales of each state as well as a probabilistic description

of the spatial relationships between states. This record is similar to a multi-point histogram

or an indicator variogram in the vertical "up"

103

direction. A simulation based on a geometric description will recapture the vertical

spatial relationships between categories if these statistics are enforced.

The cores of the Confining Unit were logged in detail as described above. From the

core logs, a record of lithofacies was prepared using a 2-mm sampling interval. A data file

was built from these records and a Fortran computer program was written to read the data

file and calculate the transition matrix and its properties. All missing section in each core

was assigned to the bottom of the core. Transitions between cores were not counted.

The original records were collected on a 2-mm vertical spacing as this was the smallest

interval for which most visible vertical variability could practically be recorded. The pitfall

of using too small of a sampling interval is that it can lead to diagonally dominant

transition matrices. Choose too large of a sampling interval and significant transitions may

be: missed. Since the choice of sample spacing is somewhat arbitrary, Schwarzacher (1975)

discussed how an optimal minimal sample spacing could be chosen by trying a range of

possible values on a trial section and choosing the spacing that results in a matrix that,

when simulated, best reproduces the statistical qualities of the trial section. The sample

spacing issue becomes less of a concern if embedded forms are used as part of the Markov

analysis.

The tally or transition frequency matrix for the Corifining Layer at a sample spacing

(A) of 2 mm is shown in Table 5.3.

104

State Med. Sand Fine Sand Silt Silty

Clay Clay Diamict


Silt 5 22 656 16 3 3 Silty Clay 1 90 22 2938 7 10

Clay 1 12 3 13 640 0 Diamict 4 11 1 7 1 787

Table 5.3: Single-dependent transition frequency matrix for Gloucester Confining Layer

The corresponding transition probability matrix for a sample spacing of 2-mm for the

Gloucester Confining Layer is shown in Table 5.4.

State Med. Sand Fine Sand Silt Silty Clay

Clay Diamict

Med. Sand 0.9754 0.0133 0.0066 0.0022 0.0000 0.0022 Fine Sand 0.0009 0.9327 0.0119 0.0414 0.0085 0.0042

Silt 0.0070 0.0312 0.9305 0.0227 0.0042 0.0042 Silty Clay 0.0003 0.0293 0.0071 0.9576 0.0022 0.0032

Clay 0.0014 0.0179 0.0044 0.0194 0.9566 0.0000 Diamict 0.0049 0.0135 0.0012 0.0086 0.0012 0.9704

Table 5.4: Single-dependent Markov transition matrix for Gloucester Confining Layer.

The marginal probability vector of the data obtained by powering the matrix until the

column vectors stabilize was found to be comparable to the observed proportions of each

facies (Table 5.5).

105

State Med Sand

Fine Sand

Silt Silty Clay Clay Diamict

Observed 0.058 0.269 0.091 0.393 0.086 0.104 Calculatd 0.069 0.268 0.099 0.378 0.086 0.100

Table 5.5: Comparison of observed proportion of each lithotype in Confining layer with marginal probability vector calculated from Markov transition matrix.

The calculated %2 test statistic is 33607 for 25 degrees of freedom under the

ass;umption that rare events can be included. The critical value at a 95% confidence level

is 37.65. The hypothesis that the observed Markov chain is equivalent to an independent

series is thus soundly rejected. In actuality, this is a trivial result. The test statistic is large

because the transition matrix is so diagonally dominant. The same series of observations

also show a strong double dependence for any number of combinations of reasonable lags,

another trivial result due to the diagonal dominance.

Close inspection of the geological records of the Confining Layer and the histograms

of bed thickness suggested there may be least two scales of heteorgeniety. Most of the

bedding occurs on the scale of 10 to 50 mm. Within these beds, however, there are often

fine interbeds at the scale of 2-6 mm. Most of these beds are thin fine sand or silt

interbeds in clay or silty clay. The geological significance of these thin bedded events is

uncertain. They may be very short-lived, minor turbidity events on the submerged delta

aprons or are related to storm surges below the fair-weather wave base. Regardless of

their origin, these small scale, localized features will not significantly affect fluid flow or

transport in the Confining Unit but may bias the Markov chain analysis.

To examine the effects of inclusion or omission of recording these fine-scale events,

the Gloucester core data set was "filtered". All beds less than 6 mm (3 unit lags) were

assigned the state of the encasing lithology. For example, all fine sands found in clay were

re-assigned a state of clay. When thin beds occurred at the

106

boundaries between thick contrasting beds, the state was re-assigned whichever of the

overlying or underlying bed had a similar grain-size. The Markov analysis was then

repeated. The Markov geometric transition frequency matrix for the filtered data using a

unit lag of 8 mm is shown in Table 5.6.

State Med. Sand Fine Sand Silt SiltyClay Clay Diamict Med. Sand 102 4 3 1 0 1 Fine Sand 2 435 11 40 6 8

Silt 4 8 141 12 2 2 Silty Clay 1 39 19 708 7 9

Clay 0 5 1 10 151 1 Diamict 4 10 0 6 2 180

Table 5.6: Transition frequency matrix for filtered data from the Confining Layer.

The corresponding transition probability matrix for the filtered data using a unit lag of

8 mm is shown in Table 5.7.

State Med. Sand Fine Sand Silt SiltyClay Clay Diamict Med. Sand 0.9189 0.0360 0.0270 0.0090 0.0000 0.0090 Fine Sand 0.0040 0.8665 0.0219 0.0797 0.0120 0.0159

Silt 0.0237 0.0473 0.8343 0.0710 0.0118 0.0118 Silty Clay 0.0013 0.0498 0.0243 0.9042 0.0089 0.0115

Clay 0.0000 0.0298 0.0060 0.0595 0.8988 0.0060 Diamict 0.0198 0.0495 0.0000 0.0297 0.0099 0.8911

Table 5.7: Transition probability matrix for filtered data from the Confining Layer.

The x2 test statistic for the transition probability matrix of the filtered data set is

6851.2. Again, this matrix is significantly different from an independent series of events

because of the diagonal dominance of the transition matrix. Since the value of %2

decreased, this experiment suggested that inclusion of small-scale bedding did make a

contribution to the detected Markovian structure.

107

A temporal reseating was also attempted to see if a Markov structure was being

masked by using equal thickness measurements instead of equal time measurements.

Schwarzacher (1975) demonstrated how a Markov analysis of geologic process derived

from equal thickness measurements of geologic facies can mask information about time

processes if variations in depositional rates between facies are not considered. The details

of conversion from equal space to an equal-time spacing are given in Chapter 4.

There are no bedding planes, datable marker beds or fossils within the Confining Layer

wiiich can be used to date horizons or derive site-specific depositional rates of the

lithotypes. For the purpose of this experiment, hypothetical depositional rates were used

(Table 5.8). These rates are comparable to depositional rates derived for analogous

Holocene glaciomarine deposits in the Canadian High Arctic. Such modern deposits exist

in a present periglacial landscape and are believed to represent conditions similar to

ancient deposits in southern Canada (Hein and Mudie, 1991).

State Relative

Depositional Rate (mm/ka)

Med. Sand 1 Fine Sand 1

Silt 0.2 Silty Clay 0.1

Clay 0.1 Diamict 1

Table 5.8: Hypothetical relative depositional rates for lithotypes.

After temporal reseating, the transition frequency matrix is as shown in Table 5.9.

108

State Med. Sand Fine Sand Silt Silty Clay Clay Diamict Med. Sand 437 6 3 1 0 1 Fine Sand 2 1957 25 87 18 9

Silt 5 22 3280 16 3 3 Silty Clay 1 90 22 29380 7 10

Clay 1 12 3 13 6400 0 Diamict 4 11 1 7 1 787

Table 5.9: Markov transition frequency matrix after temporal rescaling with hypothetical depositional rates in Table 5.8.

The rescaled transition probability matrix is shown in Table 5.10.

State Med. Sand Fine Sand Silt SiltyClay Clay Diamict Med. Sand 0.9754 0.0133 0.0066 0.0022 0.0000 0.0022 Fine Sand 0.0009 0.9327 0.0119 0.0414 0.0085 0.0042

Silt 0.0070 0.0312 0.9305 0.0227 0.0042 0.0042 Silty Clay 0.0003 0.0293 0.0071 0.9576 0.0022 0.0032

Clay 0.0014 0.0179 0.0044 0.0194 0.9566 0.0000 Diamict 0.0049 0.0135 0.0012 0.0086 0.0012 0.9704

Table 5.10: Markov transition matrix for Gloucester Confining Layer after temporal rescaling.

As with the geometric Markov chain, this matrix is significantly different from an

independent series of events based on a x2 test. Again, this is a trivial result due to the

diagonal dominance of the matrix.

The calculated marginal probability vector for the temporal Markov chain is compared

to the calculated geometric marginal probability vector in Table 5.11. One way of reading

this table is to say that a sufficiently long realization of the stochastic process would be in

'silty clay' 68% of the time but this state would only occupy 38% of the total space

occupied by the chain.

109

State Med.Sand Fine Sand Silt Silty

Clay Clay Diamict

Temporally Rescaled

0.013 0.050 0.087 0.679 0.153 0.019

Conventional

0.069 0.268 0.099 0.378 0.086 0.100

Table 5.11: Comparison of marginal probability vectors calculated from temporally-rescaled and conventional Markov transition matrices.

Length Scale Information

The conventional Markov analysis revealed a strong-first order dependency related to

the fact that the lithotypes occur in bedding. It is interesting to compare the observed

bedding thickness information with the length scale properties implicit in the transition

probability matrix. The distribution of measured thicknesses by lithotype are shown in

Figure 5.7.

The distribution of thickness or duration of any state in a simple Markov processes

will be geometric and have an arithmetic mean approximated by

Pa (1-p/i)

A (5.4)

where A is the unit step length and pn is the state-to-same-state transition probability.

Krumbein (1975) noted that in empirical stratigraphic studies, sequences occur that have a

significant Markov property in their embedded form but do not have a geometric

distribution of bed thicknesses. Rather, they may possess lognormal, truncated normal, or

similar distributions (e.g. Schwarzacher,1975).

110

Unit Thickness (mm) Unit Thickness (mm)

Unit Thickness (mm)

1 30

o O

£ 3

20

10

Silty Clay

• n_r-inr-50 100

Unit Thickness (mm)

Unit Thickness (mm)

. 0

o O

£ 3

20

10

Diamict

nnrnnrim 50

Unit Thickness (mm) 100

Figure 5.7: Thickness distributions of "beds" or units by lithotype.

I l l

The arithmetic means of observed lithotype thicknesses, b, are compared to the

expected arithmetic mean of the geometric distribution, E(b)=pi;/( 1 -pii) A , for a series of

step sizes in Table 5.12. The reason for trying various step sizes is to see if there is an

optimal step size that replicates the observed bed thickness behaviour as mentioned above.

State Mean b (mm)

E(b): A=2 mm

E(b): A=4 mm

E(b): A=6 mm

E(b): A=8 mm

E(b): A=10 mm

Med Sand

45.60 79.45 85.20 74.73 90.67 78.00

Fine Sand

14.45 27.76 32.00 35.88 37.96 39.40

Silt 24.41 26.78 31.30 32.00 33.94 37.00 Silty Clay 31.39 45.20 48.34 57.09 58.35 61.16

Clay 24.59 44.14 52.00 50.00 60.00 67.06 Diamict 33.39 65.58 63.17 63.65 61.91 61.36

Table 5.12: Comparison of arithmetic mean of bed thicknesses with theoretical expected value of an underlying geometric distribution for a sequence of sample lags.

The mean measured values are less than the expected value if a well-behaved

geometric Markov chain is assumed. This discrepancy may be due to a censored sample

(thick values of each state are rare events and not observed) or a biased matrix in terms of

too many zero off-diagonal elements (not enough observations of rare events). The

discrepancy could also indicate that the Confining Layer is one of Krumbein's exceptions

to geometric distributions of bed thicknesses.

Another measure of central tendency in state thickness is the median body influence

length or X5o (Rosen and Gustafson, 1996). The diagonal elements of a Markov matrix are

the probability that a state will succeed itself in one step. If the Markov matrix is powered

n times, the elements on the diagonal will be the probability that a state will succeed itself

after n steps in the chain, allowing for transitions into

112

and out of other states along the way. If a diagonal element, p;i is powered alone n times,

the result is the probability that, given one starts in state i, the chain stayed in state i for n

steps without passing through another state along the way. The length corresponding to a

probability, C, that a process will remain in a same state is called the body influence length

for that specific probability level:

Pi, = C (5.5) or

Jog£ (5.6)

where C is a probability and A, = nA is the corresponding body influence range, A being the

unit step length. Rosen and Gustafson define A,5o, which corresponds to C=0.50, as the

median body influence range. The median body influence length correlates to the median

thickness of a Markov state in a one-dimensional chain or the median radius of bodies in a

two or three-dimensional field. The observed median thicknesses of the geometric

Markov chain are compared to the values of Xo.5o calculated from the transition matrix in

Table 5.13.

State Median b (mm)

A.0.50

A=2 mm XOJSO

A=4 mm X.0.50

A=6 mm A.0.50

A =8 mm A.0.50

A=10 mm Med. Sand 36 55.76 60.43 53.85 65.58 57.46 Fine Sand 6 19.93 23.54 26.90 28.99 30.65

Silt 10 19.24 23.05 24.20 26.20 28.97 Silty Clay 24 32.02 34.88 41.62 43.16 45.77

Clay 16 31.28 37.41 36.70 44.30 49.87 Diamict 30 46.15 45.16 46.17 45.63 45.91

Table 5.13: Comparison of simple arithmetic mean bed thickness with median body influence length for a sequence of sample lags.

113

A comparison of the two tables (5.12 and 5.13) suggests that .50 is no better a

theoretical measure of length scales of states than the theoretical expected value E(b) for

this data. Repetition of the same experiment using filtered data had the same results.

Embedded Markov Chain Analysis of the Confining Layer

Embedded forms of Markov chains can sometimes reveal structure that is masked or

distorted in conventional Markov analysis. The embedded transition probability matrix for

the Gloucester Confining Unit is in Table 5.14.

State Medium Sand

Fine Sand

Silt Silty Clay Clay Diamict

Medium Sand

0.000 0.545 0.273 0.091 0.000 0.091

Fine Sand

0.014 0.000 0.177 0.617 0.128 0.064

Silt 0.102 0.449 0.000 0.327 0.061 0.061 Silty Clay 0.008 0.692 0.169 0.000 0.054 0.077

Clay 0.034 0.414 0.103 0.448 0.000 0.000 Diamict 0.167 0.458 0.042 0.292 0.042 0.000

Table 5.14: Upward embedded transition matrix for Gloucester Confining Layer.

A x2 test-statistic can be calculated to test for independence of states. The marginal

probability vector of the embedded chain was calculated with the iterative scheme of Davis

(1986). The calculated %2 statistic is 5.65 for 19 degrees of freedom. The test statistic is

less than the tabulated critical value of X2 of 30.14 for a confidence level of 95%. Thus

there is no reason to reject the hypothesis that the embedded chain

114

is the same as an independent series of events. This conclusion is consistent with Rust's

mass-wasting model of bed formation in these deposits.

A test statistic can be calculated for each element to see if they are individually

different than an independent series of events. This test is sometimes used to isolate

significant non-random transitions in embedded Markov chains with a high amount of

statistical noise (Turk, 1979). The test statistic is:

The test statistic follows a standard normal distribution. Thus any individual association

greater than 2.0 can be considered to be unlikely to occur 97.5% of the time in an

independent series of events (Powers and Easterling, 1982). An association accompanied

by a test statistic less than 2.0 indicates that there is a lack of association unlikely to occur

97.5% of the time in an independent series of events.

State Medium Sand

Fine Sand

Silt Silty Clay

Clay Diamict

Med. Sand 0.00 0.51 1.88 -1.49 -0.78 0.72 Fine Sand -1.30 0.00 0.31 -0.06 1.29 -0.58

Silt 3.81 -0.31 0.00 -0.72 0.05 0.40 Silty Clay -1.49 0.26 0.65 0.00 -1.17 0.38

Clay 0.51 -0.36 0.05 0.66 0.00 -1.16 Diamict 4.99 0.03 -0.90 -0.63 -0.30 0.00

Table 5.15: Values of Turk's test statistic for significance of elements in Markov transition matrix.

The calculated values of Turk's test statistic are in Table 5.15. Those values that are

greater or less than 2.00 are in bold. The only significant pattern of associations are the

transitions silt to medium sand and diamict to medium sand.

115

The geological meaning of these associations is not clear. One possible reason for the

association of diamict and medium sand may be that the melting events that deposited the

former somehow triggered the mass wasting events suggested by Rust (1977) to have

deposited the latter. The observation that contacts in core between diamict and medium

sand tend to be gradational supports this conclusion.

To father investigate the statistical behaviour of the lithotypes, a substitutability test

was performed. The details of the calculation are found in Davis (1986). If two states

have a high degree of substitutability, it means that they share the same associations with

other states. A high degree of substitutability may mean that two identified states are

really one or that two states have a natural grouping unrecognized by other criteria. In the

substitutability matrix, the closer a value is to 1.00, the higher the degree of mutual

substitutability. A value of 0.9 is considered to be quite high. The results are shown in

Table 5.16.

State Medium Sand

Fine Sand Silt Silty Clay

Clay Diamict

Med.Sand 1.000 0.220 0.071 0.118 0.171 0.104 Fine Sand 0.220 1.000 0.373 0.006 0.278 0.400

Silt 0.071 0.373 1.000 0.583 0.879 0.954 Silty Clay 0.118 0.006 0.583 1.000 0.631 0.558

Clay 0.171 0.278 0.879 0.631 1.000 0.856 Diamict 0.104 0.400 0.954 0.558 0.856 1.000

Table 5.16: Substitutability matrix for Gloucester Confining Layer.

The substitutability analysis suggests a natural grouping of silt and clay lithotypes but

not a substitutability of either with the silty clay. This suggests that there may be two

lithotypes in the group "silty clay" indistinguishable on the basis of visual inspection alone:

one associated with gradations between silt and clay and another generated by another

process.

116

The high degree of substitutability between diamict and silt probably stems from the

significant upward association between these two lithotypes and medium sand because

there is no detectable association between diamict and silt. The high substitutability of

diamict and clay is then perhaps a consequence of the high substitutability between clay

and silt and between diamict and silt.

A test for directionality was performed. Directionality is also called asymmetry or

irreversibility in Markov chains (see Chapter 3). A %2 statistic can compare the upward

embedded matrix to the downward embedded matrix (see details in Powers and

Easterling, 1982). The test statistic was calculated to be 12.22 for 15 degrees of freedom.

The tabulated chi-squared value is 24.99 for 15 degrees of freedom at a 95% confidence

level. Since the test statistic is less than this value, there is no reason to reject the

hypothesis that the entire downward embedded matrix is the same as the upward matrix.

The eigenvectors for both regular conventional transition matrix and the embedded

transition matrix were calculated. No complex eigenvalues were found so there is no

evidence of cyclicity in the embedded matrix (Schwarzacher, 1975).

An embedded transition matrix was also calculated for the filtered data set. The main

difference between the filtered and unfiltered embedded matrix is that the fine sand to clay

transition is no longer significant. The filtering removed most of the fine sand interbeds in

clay. Other than that, the embedded Markov transition matrix after filtering is no more or

less structured than prior to filtering.

117

Conductivity and Porosity of the Gloucester Confining Layer

Hydraulic conductivity and porosity measurements were made on the sediment cores.

The goal was to characterize the hydraulic parameters of each lithotype comprising the

Confining Unit. Such characterization allows the mapping of hydraulic characteristics

onto a geosystem model for flow and transport simulation.

All of the cores from the Gloucester Waste Disposal Site were subsampled in ~10 cm

lengths. The physical tests were performed before the cores were extruded from the core

sleeves. But because of smear on the core sleeves, it was impossible to describe lithology

before extrusion. Therefore a large number of mixed lithology measurements had to be

made in order to gain a small sample population of pure lithotypes.

The vertical hydraulic conductivity of each core was measured in a falling head

permeameter (e.g., deMarsily, 1986). When possible, the hydraulic conductivity was also

measured with a vertical load that approximated in situ stress conditions. Loads were

calculated assuming a water table at 3 m below ground, a grain density of 2500 kg/m3,

and 30 percent porosity. Actual porosity was measured by dividing the difference

between saturated weight and dry weight by the core subsample volume.

Hydraulic Conductivity

The vertical loading did not change the vertical hydraulic conductivity significantly (Figure

5.8). As a consequence, the larger sample population of unloaded results are used in this

analysis. The lack of response to loading suggests the sediments are overconsolidated

with respect to their present depth of burial. The lack of substantial vertical length

changes in the cores when loaded further substantiates an overconsolidated state.

118

-8.5 -8 -7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4

Log10 of Falling Head Hydraulic Conductivity (m/s) Unloaded.

Figure 5.8: Scatter plot of measured K (no load) versus K under a load approximating subsurface total vertical stress.

119

A histogram of the logarithm of vertical hydraulic conductivity (Kv) measurements is

in Figure 5.9. The distribution of all values of logio Kv is somewhat bell-shaped. The pure

lithotype cores (identified from visual description) were extracted from the sample

population. Their logarithmic histograms are in Figure 5.10. In the case of fine sand, the

distribution of logio Kv is somewhat bell-shaped. The distribution of Kv in silty clay may

be bimodal, which would support the suspicion that there are two different subpopulations

of silty clay. In the other lithotypes, the number of pure samples is too small to make any

statements about their distributions.

The spatial structure of the hydraulic conductivity values was explored with simple

variograms. The hydraulic conductivities are measured on a sample support about ~10 cm

vertical by 3.5 cm diameter. The vertical variogram built by combining all experimental

values of logioKv under the assumption of stationarity is in Figure 5.11. The variogram

was constructed using the GSLIB program gamv3 (Deutsch and Journel, 1992). A model

vertical variogram with a sill of 0.75, a range of 0.5 m, and a nugget of 0.50 appears on

the: figure for comparison. The existence of a nugget is not surprising since the core

volumes include beds of variable lithology on a smaller scale. The range of the variogram

is about an order of magnitude greater than the mean bedding bed thicknesses summarized

in Table 5.12. This suggests there may be another scale of heterogeneity in the Confining

Layer sediment that is not being captured by the Markov statistics reported here.

The horizontal variogram of vertical hydraulic conductivity is in Figure 5.12. There is

no structure in the horizontal variogram. The lack of structure indicates that the

horizontal length scale of vertical hydraulic conductivity of the Confining Layer is less

than the average spacing between the boreholes.

301 1 : 1 1

25 -

20 -

>-. o a <D 3

cr .8 15 -

10 -

5 -

0I 1—U—U—II II—U—U—LI—u—U—I -9 -8 -7 -6 -5 -4 -3

Log10 of measured K (m/s)

Figure 5.9: Histogram of log10 (unloaded KJ measurements.

121

10

S3

u §- 5

0 -9

10

a 5 a- J

Medium Sand

rrrrrJI -8 -7 -6 -5 -4

Log10 K,

Silt

n rm -8 -7 -6 -5 -4

Log10 K,

10

o c ID

o

10

C u -5 5 o t - l

Silty Clay

rfVi nrnJI XL

Diamict

rfl n FVn -8 -7 -6 -5

Log10 K,

-8 -7 -6 -5 -4

Log10 K

-4

Figure 5.10: Histograms of log10 of ¥^ (m/s) for cores of pure lithotypes.

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8

Lag (m)

Figure 5.11: Variogram of log]0 of measured K,. All measurements assigned to midpoint of core. A model spherical variogram (dashed) with a nugget of 0.5, a sill of 0.75, and a range of 0.5 m is shown for comparison.

123

Figure 5.12: Exploratory horizontal variogram of log10 of measured K .

Porosity

124

Porosity of cores was determined by weighing the saturated cores within their sleeves

and measuring their length and diameter prior to extrusion. After extrusion, the sleeve

was weighed. The cores were thoroughly dried in a sample oven and reweighed. The

porosity was calculated as the volume of water lost, determined by weight loss after

drying, divided by the core volume. Some loss of the core volume was unavoidable during

extrusion because the sediment smeared along the insides of the core sleeves. Any loss of

core volume results in overestimation of porosity. Therefore these values can be regarded

as maximum values. Calculated porosities over 0.50 are especially suspect.

The average porosity of cores containing a pure lithotype are presented in Figure 5.13.

All lithotypes have porosities in the range of 0.30-0.50. The lowest values are found in

diamict cores which is expected since they are the most poorly sorted. The values of

porosity are cross-plotted against the logarithm of vertical hydraulic conductivity in Figure

5.14. No pattern is evident that would allow hydraulic conductivity prediction from

porosity data. No pattern is evident within the six lithotypes either.

Discussion

The main objective of this chapter was to characterize vertical variability of the layers

within the Confining Layer using Markov analysis. The Markov analysis reveals that

overall there is a great degree of randomness in the vertical interrelationships of lithotypes

in the deposit. The Markov analysis suggests possible genetic relationships between

diamict and initiation of medium-sand deposition. Vertical directionality or cyclicity

related to these hidden associations could not be detected in the embedded

U

0.2

Medium Sand

nTTI n 0.4 Porosity

0.6

15

10 c a a " 5

Fine Sand

11 n 0.2 0.4

Porosity 0.6

3 O

D

Silty Clay

H 0.2 0.4 0.6

Porosity

g-

n

Diamict

n nn 0.2 0.4

Porosity 0.6

Figure 5.13: Histograms of porosity by lithotype.

126

1

0.9

0.8

0.7

0.6

CO

8 0.5 o * * (X

0.4

0.2

0.1

0

X "•K 7>-R NL/ /ft S£ft /ft. W VJ

?ft ' 7tx M/ w i ) j ; w

* , # ^ * ^ If v,

0.3 h * * * *

-7 -6 -5 -4 Log,0 Vertical Hydraulic Conductivity

Figure 5.14: Porosity versus log10 of K,, (m/s) for all lithotypes.

127

Markov matrix, presumably because of the amount of random noise in the transition

matrix.

The results of this analysis are consistent with the depositional model of sediments at

the site. The depositional model is a subaqueous esker outwash-delta complex.

Deposition of sediments on such deltas is probably due to a combination of sediment

rainout as the subglacial channel empties into slack water as well as episodic turbidity

currents, possibly initiated by glacier calving. The sediment rainout will show sorting with

distance from the glacier front, modified by currents and delta-lobe switching. The only

significant upward lithotype association appears to be that of medium sand following

diamict, which suggests a genetic relationship of the medium sand to the events depositing

the diamict. Episodic events related to ice-wasting complicate the structure and no doubt

contribute to the high degree of apparent randomness in the bedding structure.

It was hoped that the measurements of hydraulic conductivity would show better

correlation with the six lithotypes than they did. If hydraulic conductivity were better

defined, then categories generated in a stochastic simulation could be assigned modal or

mean values of K by lithotype for flow modeling. The large variance in Kv even in pure

lithotype cores pre-empts this type of analysis. Some of this variance may be due to

leakage down the sides of the core sleeves or through unseen vertical fractures.

Representative values of K may in future be better obtained by calculation from grain-size

analyses.

The lack of clear success in characterizing complex heterogeneity in this case using

Markov measures of vertical variability should not discourage others from attempting a

similar course of investigation. The Markov structures actually correspond to the

geological model of the deposit, that being one with some type of autocyclic structure

with a large amount of superimposed noise from glacial wasting. That there is any

128

correspondence at all in such a complex unit as the Gloucester Confining Layer should

perhaps even encourage others to try this approach with more "orderly" deposits.

129


Agterberg, F.P., 1974. Geomathematics - Mathematical Background and Geo-Science

Applications. Elsevier, New York, 596 pp.

Davis, J.C., 1986. Statistics and Data Analysis in Geology. 2nd Edition. J.Wiley & Sons,

New York, 646 pp.

deMarsily, G., 1986. Quantitative Hydro geology. Academic Press, New York, 440 pp.

Deutsch, C.V., and A. JourneL 1992. GSLIB Geostatistical Software Library and User's


Doveton, J.H., 1994. Theory and Application of Vertical Variability Measures from

Markov Chain Analysis. In: Yarns, J.M., and R.L. Chambers, eds. Stochastic Modeling

and Geostatistics - Principles. Methods and Case Studies. AAPG Computer Applications

in Geology, No. 3. American Association of Petroleum Geologists, Tulsa, Oklahoma, p.

55-64.

French, H.M., and B.R. Rust, 1981. Stratigraphic Investigation - South Gloucester

Special Waste Disposal Site. Unpublished consultant report to National Hydrological

Research Institute, Environment Canada, Contract OSU80-00313.

Fulton, R.J., T.W. Anderson, N.R. Gadd, C.R. Harington, I.M. Kettles, S.H. Richard,

C.G. Rodrigues, B.R. Rust, W.W. Shilts, 1987. Summary of the Quaternary of the

130

Ottawa Region. In: R.J. Fulton, ed., Quaternary of the Ottawa Region and Guides for Day

Excursions. XIIINQUA Congress, July 31-August 8,1987. p. 7-22.




Geologic Testing Consultants, 1983. Hydrostratigraphic Interpretation and Ground Water

Flow Modeling of Gloucester Special Waste Site. Unpublishied consultant report to

National Hydrology Research Institute, Inland Waters Directorate, Environment Canada.


Wiley & Sons, Toronto, 575 pp.

Hein. F.J., and P.J. Mudie, 1990. Glacial-marine sedimentation, Canadian Polar Margin,

north of Axel Heiberg Island. Geological Survey of Canada Contribution 89078, Ice

Island Publication 21.

Jackson. R.E., R.J. Patterson, B.W. Graham J- Bahr, D. Belanger, J. Lockwood, and M.

Priddle. 1985. Contaminant Hydrogeology of Toxic Organic Chemicals at a Disposal

Site. Gloucester, Ontario. 1. Chemical Concepts and Site Assessment. NHRI Paper No.

23. Inland Waters Directorate Scientific Series No. 141. Environment Canada, Ottawa,

Canada. 114 pp.

Jackson. R.E., S. Lesage, M.W. Priddle, A.S. Crowe, and S. Shikaze, 1991. Contaminant


Remedial Investigation. Inland Waters Directorate Scientific

131

Series No. 181. National Water Research Institute, Environment Canada, Burlington,

Ontario. 68 pp.

Kao, E.P.C., 1996. An Introduction to Stochastic Process. Duxbury Press, Toronto, 438

pp.

Krumbein, W.C., 1975. Markov models in the earth sciences. In: R.B. McCammon, ed.,

Concepts in Geostatistics. Springer Verlag, New York, 168 pp.

Miall, A.D., 1985. Architectural-element analysis; a new method of facies analysis applied

to fluvial deposits. Earth Science Reviews Vol 22, no.4, p. 261-208.

Powers, D.W., and R.G. Easterling, 1982. Improved methodology for using embedded

Markov chains to describe cyclical sediments. Journal of Sedimentary Petrology, Vol. 52,

no. 3, p. 913-923.


estimation of hydrogeological properties. Ground Water, Vol. 34 no. 5, p. 865-875.

Rust, B.R., 1977. Mass flow deposits in a Quaternary succession near Ottawa, Canada;

diagnostic criteria for subaqueous outwash. Canadian Journal of Earth Science, Vol. 14,

p. 175-184.

Rust, B.R., and R. Romanelli, 1975. Late Quaternary subaqeuous outwash deposits near

Ottawa, Canada. In: Jopling, A.V., and B.C. McDonald, eds., Glaciofluvial and

Glacioulacustrine Sedimentation. SEPM Special Publication No. 23. Society of

Economic Paleontologists and Mineralogists, Tulsa, Oklahoma, p. 177-192.

132

Schwarzacher, W., 1975. Sedimentation Models and Quantitative Stratigraphy. Elsevier,

New York, 383 pp.

Turk, G., 1979. Transition analysis of structural sequences: Discussion. Geological

Society of America Bulletin, Part I, vol. 90, p. 989-991.

Walker, R.G., 1979. Facies and Facies Models 1. General Introduction. In: R.G. Walker,

ed., Facies Models, 1st Edition. Geological Society of Canada Reprint Series 1, p. 1-7.

133

Chapter 6

The Performance of Simulated Annealing With Respect to Stochastic

Reconstruction of Heterogeneity from Markov Statistics

In the prior chapters, it was shown that Markov fields can be built by simulated

annealing and that these fields can capture geologically meaningful relationships. A field

study provided an example of how the Markov structure can be derived from typical field

data. The objective of this chapter is to document practical issues that affect the

stochastic reconstruction of Markov fields by simulated annealing. As part of this

demonstration, 2D fields are constructed from the vertical variability measures of the

Gloucester Confining Layer reported on in Chapter 5. These fields are unconditional

because the computer resources needed to make a fine-enough grid to condition the

realizations to core data are not available.

The work of this chapter shows that if simulated annealing is used to build Markov

fields, then the following issues need to be considered:

> Cooling schedules need to be optimized to reduce annealing times.

> Annealing performance degrades as length scales increase relative to domain size.

> Iterative improvement can be used to speed convergence.

> Enforcing only a single-dependency Markov structure appears insufficient to build

complex Markov fields. Annealing can find an optimum solution that satisfies the

transition matrix exactly but does not replicate the length-scale information embedded

in the same matrix.

The remainder of this chapter first documents my implementation of simulated

annealing to build Markov fields with emphasis on the form of the objective function.

Then I provide experimental evidence to support the above generalizations.

134

Implementation and the Form of the Objective Function

One key to implementing simulated annealing to solve any problem is formulation of

an objective function in a way that can be updated very quickly. As in any optimization

problem, the objective function in annealing can be thought of as an n-dimensional surface.

The optimization problem can be thought of as a search on the surface for the lowest spot,

or global minimum. Depending upon the nature of the problem the surface of the

objective function can vary from simple and smooth to pathologically "lumpy". Simulated

annealing is well-suited for the latter type of problem because it is a "liill-climbing"

method when implemented in its true form. Early in an annealing run, the probability of

accepting a perturbation that increases the value of the objective function is relatively high

and the search for the global minimum does not easily get trapped in shallow local rninima.

Defining a satisfactory objective function is the first step in annealing. Following the

work of Deutsch (1992), the objective function that I employ in my annealing program

takes a flexible form that can be rapidly updated following a perturbation.

The general form of the objective function for three dimensions is:

1 3 im \dtr?in —dre,al~\2

0 = -^{L L[wi -r- ] ^ idir=\ i=\ aidir,i

3 im im \odt??in--odre?l-'\2

, V V V r Luuidir,ij uuidir,ij\

+ L la L [W2 -1Z- ] (6.1)

idir=\ i=lj=lj±i Od

im

+ TJw3[pfain-p?al]2} i=l

train idir,ij

135

where O is the value of the objective function, idir is the number of orthogonal directions

to enforce the Markov structure, im is the number of states, (f"n is the value of number of

transitions on the diagonal of the model or training Markov transition frequency matrix,

deal is the number of transitions on the diagonal of the Markov transition frequency matrix

calculated for the image being annealed, od stands for the values of the off-diagonal

elements of the Markov transition frequency matrix, and p is the proportion of each state.

The calculated value of the objective function is normalized by the initial value of O, O0.

Unequal weighting of diagonal versus off-diagonal element can be implemented by making

the values of wi and W2 unequal. The fast updating scheme is that provided by Deutsch

and Journel (1992) wherein only the contributions of the perturbed nodes to the

multi-point histograms are updated during annealing, eliminating the need to recalculate

the entire objective function at each step.

The third term is a penalty term used to penalize departures from the global histogram

of states with weight wj. This term may be applied if perturbation is done by drawing

nodal values from the underlying population histograms (as used by Deutsch and Journel,

1992) rather than by swapping randomly chosen pairs of nodes in a field initially seeded

with the correct proportions of each state. Experiments suggested that conventional

annealing using pair swapping was more effective in reaching threshold minimum O-values

for Markov fields than the method of perturbing single nodes by drawing values from the

underlying global histogram, even when the penalty term was enforced in Equation 6.1.

Neighbourhood reduction (excursion limiting) in annealing, whereby the neighbourhood

for pair swapping is reduced with progress of the annealing, was not tried.

The values of squared differences between training and real images are normalized by

the training value to give more equal weighting to small values in the Markov

136

transition structure. If the training value is zero, the squared difference is divided by one

instead.

Effects of the Cooling Schedule

An efficient cooling schedule is an integral part of using simulated annealing. The

results of my experiments only underscore this observation already reported by others.

The cooling schedule is an empirical balance between computational time and the

unknown morphology of the objective function. If the cooling proceeds too quickly, the

annealing can become "frozen" in a suboptimal state or local minimum. If the cooling

proceeds too slowly, the annealing can proceed through too many perturbations to be

effective. Deutsch and Journel (1992) propose that a standard annealing schedule be used

as a starting point for determining a cooling schedule. In their implementation, the

original value of the objective function is normalized to a value of 1.000. Dougherty and

Marryott (1992) examine the relative effects of altering annealing parameters on annealing

efficiency.

A minimum number of perturbations, kaCcePt, is defined as the number of perturbations

of a field which either successfully lower the objective function or are kept if they raise the

objective function with a probability Paccept that follows a Boltzman distribution (Jensen et

al., 1997). If a predetermined number of perturbations, kmaximwn, are reached at a given

temperature before kaCcepx is met, the annealing is stopped. Either the objective function

will be smaller than a predetermined threshold or the annealing will be considered to be

unsuccessful. As an initial guess, kaCcePt is set at 10 times the number of grid nodes and

kmaximwn is set at 10 time kaccept- The factor (A.) by which the temperature parameter, T, in

the Boltzman distribution is reduced if kacCept is reached before kmaximwn, may be set initially

to 0.1. The value of A, can be increased if the annealing run becomes trapped too quickly

in a suboptimal state.

137

Ouenes and Baghavan (1994) suggest that trial and error can identify opportunities to

economize on the cooling schedule in true annealing. A graph of the value of objective

function versus temperature is useful. Typically one would want to see the objective

function being lowered immediately after a temperature reduction. Eventually the

objective function will become asymptotic to some value. Perturbations after this point

will no longer reduce the objective function and will be inconsequential. The number of

iterations it takes to reach this asymptotic value are not known a priori, but can be found

in trial runs.

Likewise, the value of k can be optimized through trial and error. Values of k closer

to 1 will reduce the chance that the objective function will fall into local minimum but at

the cost of many more perturbations. Values of A, smaller than 0.1 may be effective if the

objective function surface is not lumpy. Again, trial and error is necessary to identify an

acceptable value of k. Cooling schedules are provided to the program manneal through a

parameter file.

Annealing can be implemented without the hill-climbing aspect. In one such

implementation, called iterative improvement, only perturbations that reduce the objective

function are accepted. Another variant is called steepest descent. In steepest-descent, all

perturbations within a predefined neighbourhood are evaluated and the perturbation which

leads to the greatest reduction in the value of the objective function is accepted. In these

variants, a cooling schedule is not required.

To illustrate the effect of cooling schedule on annealing performance, a simple

three-state categorical Markov field was annealed. The Markov chain model used is in

Table 6.1.

138

P[.j(x+l)|i(x)] j=State 1 State 2 State 3 i=State 1 0.60 0.20 0.20 State 2 0.20 0.60 0.20 State 3 0.20 0.20 0.60

Table 6.1: A simple three-state Markov categorical field.

All states have equal proportions and the same length scale. There is no structure in the

relationships between categories.

In all cases, the normalized objective function reached the preset minimum value of

le-5. The standard annealing schedule and three variants were used to anneal the field

(Table 6.2).

Trial Kaccept lniraum X A 70,000 1,000,000 0.05 B 70,000 1,000,000 0.1 C 100,000 1,000,000 0.1 D 70,000 1,000,000 0.5

Table 6.2: Four variants of the standard annealing schedule. Variant C is the standard suggested by Deutsch and Journel (1992).

Figure 6.1 is a graph of the trajectory of O, the normalized objective function, versus

number of perturbations for four different variants on the standard annealing schedule.

Tiue annealing by swapping was used with equal weighting on the diagonals and

off-diagonals of the underlying Markov structure. The initial field was seeded with the

correct proportions of each state.

In this simple example, the Markov structure contained only length scale information

with no structure in the categorical interrelationships so annealing was relatively

straightforward. However, in this example the basic tool for optimizing an annealing

schedule is made clear. If the value of the objective function did not

139

o 10

c :::

o a>

o -a u N

13 :: o 2

1.5 2 2.5 3

Number of Perturbations xlO

Figure 6.1: Comparison of objective function behaviour with different annealing schedules. The idealized structure is a simple 2D Markov field, 100x100 nodes. In A, k^^ = 70,000, X = 0.05. In B, k ^ , =70,000, =0.10. In C, k ^ =100,000, X=0.10. In D A =70,000,^=0.50.

140

stabilize so rapidly after a reduction in temperature, a plot such as shown in Figure 6.1

could identify when the asymptotic limit of the objective function is approached and the

associated number of iterations used to help define kaccept for future runs. In this particular

example, the success of Trial A, a very rapid cooling schedule, suggests that iterative

improvement may be viable. Indeed, iterative improvement does reach the same threshold

objective function in less fewer than even Trial A.

Sensitivity to Length Scales on Finite Grids

The performance of the annealing is generally poor when strongly diagonally-dominant

Markov transition structures are enforced. Diagonal dominance in Markov transition

structures results from small experimental lags being used to characterize real sequences

(see more complete discussion in Chapter 5). The diagonal elements in a Markov

transition matrix encapsulate length-scale information (see Equation 5.4). On finite grids,

the diagonal terms will determine the ratio of length scales in each category to grid-scale.

As the length-scale increases relative to the grid scale, the number of possible geometries

honouring the ideal Markov structure decreases. Consequently, it may become more

difficult for annealing to find the global minimum by random perturbations.

To demonstrate the length-scale effect, four three-state Markov structures were

annealed. These structures are shown in Table 6.3. Each structure was enforced on a

100x100 node grid using iterative improvement to a maximum of 2 million perturbations.

According to Equation 5.6, the median body influence lengths of each state in the four

structures are, respectively: 2.4, 4.3, 6.7, and 13.5 units. The corresponding ratios of

giid dimension to median body influence lengths are 41.5, 23.4, 15.2, and 7.4 units.

141

P{j(x+l)i(X) j=State A State B State C i=State A 0.750 0.125 0.125 State B 0.125 0.750 0.125 State C 0.125 0.125 0.750

P{j(x+l)|i(x) j=State A State B State C i=State A 0.850 0.075 0.075 State B 0.075 0.850 0.075 State C 0.075 0.075 0.850

P{.j(x+l)|i(x) j=State A State B State C i=State A 0.900 0.050 0.050 State B 0.050 0.900 0.050 State C 0.050 0.050 0.900


Table 6.3: Four simple three-state Markov transition matrices with different body influence lengths.

Figure 6.2 shows the objective-function value trajectory for each case for iterative

improvement. As the ratio of grid dimension (L) to median body-influence length (X)

decreases, the performance of the annealing degrades as measured by the slope of the

objective-function trajectory. The same degradation in performance was observed to

occur when conventional annealing (i.e., with hill-climbing) was implemented. The

antidotes for the length-scale effect are either to use a larger grid or adjust the cooling

schedule to have a higher X and larger kmaximum.

142

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Number of Perturbations x 10

Figure 6.2: Effect of length scale on annealing performance. The length scale is the body influence length, A,, defined by Equation 5.5. The domain size, L, is 100 units. The ratio of domain size to length scale is shown adajacent to each trajectory. The smaller the ratio of domain size to length scale, LA, the poorer the annealing performance. Iterative improvement is used in this demonstration.

143

Iterative Improvement or Conventional Annealing?

Neither iterative improvement nor conventional annealing were found to be

particularly successful in generating complex Markov structures - especially those with

five or more states and complicated, non-random interrelationships between categories.

Experience showed that the best annealing results for complex Markov fields were

attained when true simulated annealing with random swapping of values on a grid seeded

with prescribed proportions of each state was followed by iterative improvement using the

output of the conventional annealing.

For example, consider the three-state Markov field captured in the Markov transition

matrix in Table 6.4. There is a non-random, complex structure in the categorical

interrelationships and the length scale of state C is different from the other two.


Table 6.4: A more complicated three-state Markov transition matrix.

Figure 6.3 shows the objective-function trajectories for both iterative improvement

and conventional annealing by pair swapping. In both cases, the desired minimum value of

O could not be obtained within 2 million perturbations. However, by post-processing the

realization built by conventional annealing with iterative improvement, a satisfactory value

of the objective function could be attained in less than 2 million perturbations. The

combined approach was found to be the only way to prepare the realizations discussed

below.

O-trajectory: true anenaling by swapping. Final normalized objective function: 4.6e-08 after 1,606,352 - standard cooling schedule.

O-trajectory: iterative improvement. Final normalized objective function: 1.0e-07 achieved in 11,489 succssful swaps out of 1,560,000 trials. Objective function not lowered within single precision thereafter.

O-trajectory when true annealing output was post-processed with iterative improvement. Final normalized objective function: 1.0e-7 after 909 successful swaps in 90,000 perturbations.

0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8

Number of Perturbations xlO

Figure 6.3: Comparison of objective function (O) trajectories for a three-state Markov field (see text for details).

145

Two Dimensional Reconstruction of the

Gloucester Confining Layer by Annealing

One objective of this dissertation is to see if Markov structures can be successfully

captured in stochastic representations of real geologic units. Here the Markov statistics

reported in Chapter 5 are used in reconstruction of geological heterogeneity of the

Gloucester Confining Layer using the techniques and methods put forth throughout is this

dissertation.

A five-state, 100x100 node categorical field is shown in Figure 6.4. This field

represents an unconditional depositional dip-section through the Gloucester Confining

Layer. Figure 6.5 is a similar field but represents an unconditional depositional

strike-section. Each grid cell in the realizations is 8 cm by 8 cm, the step size at which the

Markov statistics were formulated. The details of their generation are given below. These

two fields represent the culmination of this dissertation and hopefully the launch point for

future research by geologists along the Bayesian lines of Freeze et al. (1991) and Rosen

and Gustafsen (1996).

The results of Chapter 5 showed there is a complex structure in the vertical variability

of sediments which comprise the Confining Layer at the Gloucester waste disposal site.

The structure (see Table 5.10) has a high degree of randomness but there are relationships

between categories that are statistically significant and geologically supportable. Six

categories were visually identified based on lithology: medium-coarse sand, fine sand, silt,

silty clay, clay, and diamict. Very fine beds (less than 8 mm) were filtered from the record

of vertical variability.

Efforts to anneal the six-state Markov transition structure on a 100x100 grid were not

successful. The combined effects large number of states, the complex structure, and the

diagonal dominance evident in Table 5.10 were sufficient to cause both true

;jpll;

Figure 6.4: Unconditional, isotropic stochastic reconstruction of part of the Gloucester Confining Layer: dip section. North is to right because up is equivalent to deeper water in this overall transgressive environment. Grid units are 8 mm. Lithotypes are medium-coarse sand -white, fine sand -light grey, silt -medium grey, clay/silty clay -dark grey, diamict -black.

Figure 6.5: Unconditional, isotropic stochastic reconstruction of part of the Gloucester Confining Layer: strike section.East is to right. All horizontal directionality has been removed from the Markov structure. Grid units are 8 mm. Lithotypes are medium-coarse sand -white, fine sand-light grey, silt -medium grey, clay/silty clay -dark grey, diamict -black.

148

annealing and iterative improvement to fall into suboptimal states and never reach a

satisfactory match with the idealized Markov structure. When the number of categories

was reduced by one, the resulting Markov structure could be enforced on a 100x100 grid

through a combination of true annealing by pair swapping using a slow cooling schedule

(^=0.9) and then post-processing the realization with iterative improvement. Silty clay

and clay were combined into a new category based on the geological model and the

gradational bedding relationships observed between these categories. Silt and clay could

have been combined based on the substituability analysis presented in Chapter 5, but this

act would have combined units with diferent hydraulic properties. As well, the filtered

data set was used in this case.

The difference between the dip and strike sections is that in the former, the vertical

variability model is imposed directly in the horizontal. In the latter, all directionality in the

horizontal is removed. The directionality is removed by averaging the off-diagonal terms

in the transition probability matrix prior to generating the multi-point histograms to be

imposed in the horizontal. Since facies belts or mosaics are likely to migrate along dip

(i.e., away from source), it makes geologic sense to leave any directionality, statistically

significant or not, in the realization representing the dip-section. But given the likely

environment of deposition, it is not likely that there is any directionality of facies belts or

mosaics along strike. Thus it is geologically reasonable to remove directionality from the

horizontal Markov structure being imposed in this experiment. The Markov structure for

the dip-section is in Table 6.5.

State Med. Sand Fine Sand Silt Clay Diamict Med. Sand 0.8965 0.0460 0.0345 0.0115 0.0115 Fine Sand 0.0049 0.8350 0.0271 0.1133 0.0197

Silt 0.0290 0.0580 0.7971 0.1014 0.0145 Clay 0.0013 0.0583 0.0265 0.9007 0.0132

Diamict 0.0253 0.0633 0.0000 0.0506 0.8608

Table 6.5: Markov transition matrix for filtered Gloucester data set after combining silty clay and clay lithotypes.

149

The single-step Markov transition frequencies being enforced as multi-point

histograms in the dip direction are tabulated in Table 6.6 (on Page 150 following). The

results of conventional annealing and after post-processing with iterative improvement are

compared. As one can see, the results after post-processing are excellent. The same

information for the strike section is given in Table 6.7 (on Page 151 following). Again,

the results after post-processing are excellent. The time necessary to generate one

100x100 node realization on a 100 MHz Pentium PC was just over two hours.

The expected length scales of each category in Table 6.5 can be calculated from

Equation 5.4 (Table 6.8) and compared with the realizations in Figure 6.4 and 6.5.

State Med. Sand Fine Sand Silt Clay Diamict Expected

mean thickness (grid units)

8.67 5.06 3.92 9.07 6.18

Table 6.8: Expected mean state thicknesses for transition probabilities in Table 6.5.

The length scales of bodies in the figures tend to be larger than the expected mean

thicknesses from the Markov transition probability matrix. The reason for this discrepancy

is unclear. It may be that the annealing is using the pixel noise to satisfy the objective

function while managing to sidestep enforcement of the length scales. Alternatively, the

mismatch of the measured and observed bed thicknesses may be a direct consequence of

using only the first-order Markov probability matrix to reconstruct the heterogeneity. As

mentioned in Chapter 5, Krumbein (1975) noted that some measured natural sequences

with significant Markov properties also have a mismatch between expected and mean bed

thicknesses. True bed-thickness distributions are often better characterized as lognormal

or truncated normal distributions. Since a first-order Markov transition matrix will

generate a geometric distributions of body lengths or bed thicknesses, it may be that more

information on

150

From State To State In Direction Markov Model

After True Annealing

After Iterative Improvement

l (0 1) 646 606 645 l (10) 646 604 643 2 (0 1) 33 52 34 2 (10) 33 47 35 3 (0 1) 25 36 26 3 (10) 25 38 26 4 (0 1) 8 13 8 4 (10) 8 16 8 5 (0 1) 8 14 8 5 (10) 8 16 9

2 1 (0 1) 13 20 13 2 1 (10) 13 21 13 2 2 (0 1) 2,164 1,998 2.164 2 2 (10) 2,164 2,008 2,164 2 3 (0 1) 70 99 70 2 3 (10) 70 97 70 2 4 (0 1) 294 395 294 2 4 (10) 294 394 294 2 5 (0 1) 51 80 51 2 5 (10) 51 72 51 3 1 (0 1) 31 41 31 3 1 (10) 31 46 33 3 2 (0 1) 62 79 62 3 2 (10) 62 78 62 3 3 (0 1) 856 781 856 3 3 (10) 856 772 855 3 4 (0 1) 109 147 109 3 4 (10) 109 156 108 3 5 (0 1) 16 26 16 3 5 (10) 16 22 16 4 1 (0 1) 6 15 6 4 1 (10) 6 10 6 4 2 (0 1) 270 361 270 4 2 (10) 270 366 269 4 3 (0 1) 123 155 122 4 3 (10) 123 166 123 4 4 (0 1) 4,175 4,012 4,175 4 4 (10) 4,175 3.993 4,176 4 5 (0 1) 61 92 62 4 5 (10) 61 100 61 5 1 (0 1) 25 39 26 5 1 (10) 25 40 26 5 2 (0 1) 62 102 62 5 2 (10) 62 93 62 5 3 (0 1) 0 3 0 5 3 (10) 0 1 0 5 4 (0 1) 49 68 49 5 4 (10) 49 76 49 5 5 (0 1) 842 766 841 5 5 (10) 842 768 841

Table 6.6: Transition frequencies (multi-point histograms) for the horizontal (0 1) and vertical (10) directions in the dip-section reconstruction of the Gloucester Confining

Layer. State 1 corresponds to medium sand, 2 to fine sand, 3 to silt, 4 to combined silty clay and clay, and 6 to diamict. Columns 5 and 6 show the multipoint histograms after

true annealing and after post-processing using iterative improvement in comparision to the Markov model in Column 4.

151

From State To State In Direction Markov Model

After True Annealing

After Iterative Improvement

i (0 1) 646 604 645 l (10) 646 611 646 2 (0 1) 23 38 24 2 (10) 33 43 34 3 (0 1) 28 37 28 3 (10) 25 37 25 4 (0 1) 7 14 7 4 (10) S 18 8 5 (0 1) 17 28 17 5 (1 0) 8 12 8

2 1 (0 1) 23 33 23 2 1 (10) 13 16 13 2 2 (0 1) 2,164 2,013 2,164 2 2 (10) 2,164 2,027 2,164 2 3 (0 1) 66 89 66 2 3 (10) 70 91 70 2 4 (0 1) 282 372 282 2 4 (10) 294 384 294 2 5 (0 1) 56 85 57 2 5 (10) 51 74 51 3 1 (0 1) 28 30 28 3 1 (10) 31 38 30 3 2 (0 1) 66 75 66 3 2 (10) 62 70 62 3 3 (0 1) 856 792 856 3 3 (10) 856 792 856 3 4 (0 1) 116 163 116 3 4 (10) 109 152 110 3 5 (0 1) 8 14 8 3 5 (10) 16 22 16 4 1 (0 1) 7 18 7 4 1 (10) 6 9 6 4 2 (0 1) 282 376 282 4 2 (10) 270 362 270 4 3 (0 1) 116 144 116 4 3 (1 0) 123 151 123 4 4 (0 1) 4,175 4,016 4,175 4 4 (10) 4,175 4,017 4,174 4 5 (0 1) 55 81 55 4 5 (10) 61 96 62 f 1 (0 1) 17 36 18 S 1 (10) 25 47 26 5 2 (0 1) 56 90 56 5 2 (10) 62 90 62 5 3 (0 1) 8 12 8

5 3 (10) 0 3 0 5 4 (0 1) 55 70 55 5 4 (10) 49 64 49 5 5 (0 1) 842 770 841

> 5 (10) 842 774 841

Table 6.7: Transition frequencies (multi-point histograms) for the horizontal (0 1) and vertical (1 0) directions in the strike-section reconstruction of the Gloucester Confining Layer. State 1 corresponds to medium sand, 2 to fine sand, 3 to silt, 4 to combined silty clay and clay, and 6 to diamict. Columns 5 and 6 show the multipoint histograms after

true annealing and after post-processing using iterative improvement in comparision to the Markov model in Column 4.

152

bed thickness distribution is needed to be incorporated in the objective function to anneal a

more satisfactory image.

153


Deutsch, C.V., 1992. Annealing techniques applied to reservoir modeling and the


University, Calif.

Deutsch, C.V., and A.G. Journel, 1992. GSLIB Geostatistical Software Library and

User's Guide. Oxford University Press, New York, 340 pp.

Dougherty, D.E., and R.A. Marryott, 1992. Markov chain length effects in groundwater

management by simulated annealing. In Fitzgibbon, W.E. and M.F. Wheeler, eds.,

Computational Methods in Geosciences. SIAM, Philidephia, p. 53-65.



28, no. 4, p. 419-435.



Ouenes, A., and S. Bhagavan, 1994. Application of simulated annealing and other global



154 Chapter 7

Conclusion

The objective of this dissertation is to answer two questions:

> Can Markov statistical structures be imposed on structured random grids of hydraulic

conductivity (K) in an effort to inject more geological realism into stochastic

simulations of aquifer heterogeneity?

> Does their inclusion make a difference to predicting flow and transport?

The answer to both questions appears to be a qualified yes.

In Chapter 2, one particular Markovian signature, cyclicity, was imposed on

unconditional continuous K fields through use of the hole-effect covariance structure.

The main effect of using the hole effect was found to be a significant reduction in the

variance in outputs of stochastic flow and transport experiments. This reduction occurs

because the wavelength of the hole-effect oscillations is extra information that constrains

the structure in addition to length-scale. Percolation experiments suggest that

enforcement of a vertical hole-effect covariance structure increases the probability that

high values of K are connected in the horizontal. This finding is of interest because

vertical cyclicity is relatively common in real sedimentary deposits which host contaminant

plumes or hydrocarbon resources. If geological evidence suggests that a hole-effect

covariance structure is or is not appropriate to include in a geosystem model, then this

information should be heeded because the choice will have an impact on predictions of

flow and transport.

In Chapter 3, Markov transition probability matrices were encoded into multi-point

histograms and 2D categorical fields were constructed with simulated annealing.

Markovian structures with a geological significance were imposed on these fields:

155

hierarchical stratigraphic memory (double dependency), directionality, and cyclicity.

Effective hydraulic conductivity calculation showed there is an effect on flow behaviour

when these different structures are imposed, though these are not first-order effects.

Markov fields can be built in multiple dimensions from vertical variability data if one

invokes Walther's Law of Facies Succession in a probabilistic sense. In Chapter 4,1 show

how this transference of information from the vertical to the horizontal should be imposed

in a temporally-rescaled framework to be true to its geological meaning. The temporal

rescaling requires some knowledge of relative rates of deposition of the different

categories and perhaps a decompaction step. A temporal-rescaled Markov field is

analogous to a Wheeler diagram, honouring the space-time-sediment volume relationships

inherent in real deposits. In this regard, such Markov fields are superior to categorical

fields generated from empirical multi-point histograms derived from outcrop photos, for

example.

Chapter 5 documents my effort to derive a Markov statistical model of vertical

variability in a complex aquitard layer at the Gloucester waste disposal site, Ontario. In

the final analysis, the layer was found to have a large component of random noise in its

Markov transition matrices. This noise almost totally obscures any geological structure.

The only signficant upward association was found to be between gradationally-bedded

diamict and medium sand, suggestive of a genetic relationship. Correlations between

lithotypes and laboratory-measured hydraulic conductivity were confounded by the large

dispersion of values within each lithotype. Even though the Markov statistics were not

elegant, they did conform to the general depositional model for the aquitard layer - that

being a coalescent, subaqueous, proglacial outwash fan showing classic, autocyclic deltaic

characteristics but being profoundly affected by mass-wasting processes probably related

to proximal glacial wasting.

156

In Chapter 6, some performance issues related to using simulated annealing to

construct Markov fields are discussed. Practical issues like the form of the objective

function, the cooling schedule, the perturbation method, and scale effects pose significant

but not insurmountable obstacles to implementing the ideas put forth in Chapters 3-5. The

issues of pixel noise, global reproduction of transition probabilities, and poor reproduction

of length scales remain serious problems to be addressed.

The results of this work do show that Markovian analysis and Markov field

construction can bring forth more geological realism into stochastic models. Despite its

shortcomings, simulated annealing is an attractive vehicle for this work because it can

accommodate geostatistical, engineering, and geophysical data in the objective function.

In this way, Markov analysis can complement the efforts of a geosystem modeller using

any of the myriad other tools presently at their disposal.

157

Complete Bibliography

Aasum, Y., and M.G. Kelkar, 1991. An application of geostatistics and fractal geometry

for reservoir characterization. SPE Formation Evaluation, March 1991, p. 11-19.

Agterberg, F.P., 1974. Geomathematics - Mathematical Background and Geo-Science

Applications. Elsevier, New York, 596 pp.

Agterberg, F.P., and I. Banerjee, 1969. Stochastic model for the deposition of varves in

glacial Lake Barlow-Ojibway, Ontario, Canada. Canadian Journal of Earth Science, 6, p.

625-652.

Armstrong, M., 1982. Common problems seen in variograms. Mathematical Geology,

vol. 16, no. 3, p. 305-313.

Ashley, G.M., and T.D. Hamilton, 1993. Fluvial response to late Quaternay climatic

fluctuations, central Kobuk Valley, northwestern Alaska. Journal of Sedimentary

Petrology, vol. 63, no. 5, p. 814-827.

Bayer, IL, 1985. Pattern Recognition Problems in Geology and Paleontology.

Springer-Verlag Lecture Notes in Earth Sciences 2, 229 pp.

Bond, G.R., and M.A. Komitz, 1984. Construction of tectonic subsidence curves for the

early Paleozoic miogeocline, southern Canadian Rocky Mountains: Implications for

subsidence mechanisms, age of breakup, and crustal thinning. Geological Society of

America Bulletin, vol. 95, p. 155-173.

158

Box, G.E.P, and G.M. Jenkins, 1976. Time Series Analysis - forecasting and control.

Holden-Day, San Francisco, 575 pp.

Carle, S.F, and G.E. Fogg, 1996. Transition probability-based indicator geostatistics.


Datta Gupta, A., L.W. Lake, and G.A.Pope. 1995. Characterizing heterogeneous

permeable media with spatial statistics and tracer data using simulated annealing.

Mathematical. Geology, vol. 27, no. 6, p. 789-806.

Davis, J.C., 1986. Statistics and Data Analysis in Geology. 2nd Edition. J.Wiley & Sons,

New York, 646 pp.

deMarsily, G., 1986. Quantitative Hvdrogeology. Academic Press, New York, 440 pp.

Desbarats, A.J., and S. Bachu, 1994. Geostatistical analysis of aquifer heterogeneity from

the core scale to the basin scale: a case study. Water Resources Research, vol. 30, no. 3,

p. 673-684.

Desbarats. A.J., and R. Dimitrikopolous, 1990. Geostatistical modeling of transmissibility

for 2D reservoir studies. SPE Formation Evaluation, vol. 5, no. 4, p. 437-443.

Desbarats. A.J., and R.M. Srivistava, 1991. Geostatistical characterization of

groundwater flow parameters in a simulated aquifer. Water Resources Research vol. 27,

no. 5. p. 687-698.

Deutsch. C.V., 1992. Annealing techniques applied to reservoir modeling and the


University, Calif.

159

Deutsch, C.V., and T.A. Hewitt, 1996. Challenges in reservoir forecasting. Mathematical

Geology, vol. 28, no. 7, p. 829-842.

Deutsch C.V. and A.G. Journel, 1992. GSLIB Geostatistical Software Library and User's


Deutsch, C.V. and P.W. Cockerham, 1994. Practical considerations in the application of

simulated annealing to stochastic simulation. Mathematical Geology, vol. 26, no. 1, p.

67-82.

Deutsch, C.V., and L. Wang, 1996. Hierarchical object-based stochastic modeling of

fluvial reservoirs. Mathematical Geology, vol. 28, no. 6, p. 857-880.

Dougherty, D.E., and R.A. Marryott, 1992. Markov chain length effects in groundwater

management by simulated annealing. In Fitzgibbon, W.E. and M.F. Wheeler, eds.,

Computational Methods in Geosciences. SI AM, Philidephia, p. 53-65.

Doveton, J.H., 1994. Theory and Application of Vertical Variability Measures from

Markov Chain Analysis. In: Yarns, J.M., and R.L. Chambers, eds., Stochastic Modeling

and Geostatistics - Principles. Methods and Case Studies. AAPG Computer Applications

in Geology, No. 3. American Association of Petroleum Geologists, Tulsa, Oklahoma, p.

55-64.

Einsele, G., W.Ricken, and A. Seilacher, 1991. Cycles and events in stratigraphy - basic

terms and concepts. In Einsele, G., W. Ricken, and A.Sielacher, eds., Cycles and Events

in Stratigraphy. Springer-Verlag, p. 1 -22.

160

Farmer, C.L., 1992. Numerical Rocks. In King, P.R., ed., Proceedings of the First

European Conference on The Mathematics of Oil Recovery. 1989. Oxford University

Press, p. 437-448.

Fischer, A.G., and D.J. Bottjer, 1991. Orbital forcing and sedimentary sequences. Journal

of Sedimentary Petrology, vol. 61, no. 7, p. 1063-1069.

Fischer, A.G., and L.T. Roberts, 1991. Cyclicity in the Green River Formation (lacustrine

Eocene) of Wyoming. Journal of Sedimentary Petrology, vol. 61, no. 7, p. 1146-1154.

Fogg, G.E., 1989. Stochastic Analysis of Aquifer Interconnectedness: Wilcox Group,

Trawick Area, East Texas. Texas Bureau of Economic Geology Report of Investigations

No. 189, Austin, 68 pp.

Freeze, R.A., and J. Cherry, 1979. Groundwater. Prentice Hall, 604 pp.

French, H.M., and B.R. Rust, 1981. Stratigraphic Investigation - South Gloucester

Special Waste Disposal Site. Unpublished consultant report to National Hydrological

Research Institute, Environment Canada, Contract OSU80-00313.

Fulton, R.J., T.W. Anderson, N.R. Gadd, C.R Harington, I.M. Kettles, S.H. Richard,

C.G. Rodrigues, B.R. Rust, W.W. Shilts, 1987. Summary of the Quaternary of the

Ottawa Region. In: R.J. Fulton, ed., Quaternary of the Ottawa Region and Guides for Day

Excursions. XIIINQUA Congress, July 31-August 8, 1987. p. 7-22.




161

Gelhar, L., 1986. Stochastic subsurface hydrology from theory to applications. Water

Resources Research, vol. 22, no. 9, p. 135S-145S.

Gelhar, L.W., and C.L. Axness, 1983. Three-dimensional stochastic analysis of

macrodispersion in aquifers. Water Resources Research, vol. 19, no. 1, p. 161-180.

Geologic Testing Consultants, 1983. Hydrostratigraphic Interpretation and Ground Water

Flow Modeling of Gloucester Special Waste Site. Unpublished consultant report to

National Hydrology Research Institute, Inland Waters Directorate, Environment Canada.

Goovaerts, P., 1996. Stochastic simulation of categorical variables using a classification

algorithm and simulated annealing. Mathematical Geology, vol. 28, no. 7, p. 909-921.

Grant, C.W., D.J. Goggin, and P.M.Harris, 1994. Outcrop analog for cyclic-shelf

reservoirs, San Andres Formation of Permain Basin: stratigraphic framework,

permeability distribution, geostatistics, and fluid-flow modelling. American Association of

Petroleum Geologists Bulletin, vol. 78, no. 1, p. 23-54.


Wiley & Sons, Toronto, 575 pp.

Hein, F.J., and R.G. Walker, 1977. Bar evolution and development of stratification in the

gravelly, braided Kicking Horse River, British Columbia. Canadian Journal of Earth

Science, p. 562-570.

Hein, F.J., and P.J. Mudie, 1990. Glacial-marine sedimentation, Canadian Polar Margin,

north of Axel Heiberg Island. Geological Survey of Canada Contribution 89078, Ice

Island Publication 21.

162

Hohn, M.E., 1988. Geostatistics and Petroleum Geology. Van Nostrand Reinhold, New

York. 264 pp.

Isaaks, E.H., and R.M. Srivistava, 1989. Applied Geostatistics. Oxford University Press,

New York, 561 pp.

Jackson, R.E., RJ. Patterson, B.W. Graham, J. Bahr, D. Belanger, J. Lockwood, and M.

Priddle, 1985. Contaminant Hydrogeology of Toxic Organic Chemicals at a Disposal

Site, Gloucester, Ontario. 1. Chemical Concepts and Site Assessment. NHRI Paper No.

23, Inland Waters Directorate Scientific Series No. 141. Environment Canada, Ottawa,

Canada. 114 pp.

Jackson, R.E., S. Lesage, M.W. Priddle, A.S. Crowe, and S. Shikaze, 1991. Contaminant


Remedial Investigation. Inland Waters Directorate Scientific Series No. 181. National

Water Research Institute, Environment Canada, Burlington, Ontario. 68 pp.



28, no. 4, p. 419-435.

Jensen, J.L., L.W. Lake, P.W.M. Corbett, and D.J. Goggin, 1997. Statistics for

Petroleum Engineers and Geoscientists. Prentice Hall, N.J., 390 pp.

Johnson, N.M., and S.J. Driess, 1989. Hydrostratigraphic interpretation using indicator

geostatistics. Water Resources Research, vol. 25, no. 12, p. 2501-2510.

Journel, A.G. and R. Froidevaux, 1982. Anisotropic hole-effect modelling. Mathematical

Geology vol. 14, no. 3, p. 217-239.

163

Journel, A.G., and Ch.J. Huijbregts, 1978. Mining Geostatistics. Academic Press, New

York, 600 pp.

Journel, A., and J. Gomez-Hernandez, 1989. Stochastic imaging of the Wilmington clastic

sequence. Society of Petroleum Engineers paper no. 19857.

Kao, E.P.C., 1996. An Introduction to Stochastic Process. Duxbury Press, Toronto, 438

pp.

King, P.R., 1990. The connectivity and conductivity of overlapping sand bodies. In: A.T.

Buller, E. Berg, O. Hjelmeland, J. Kleppe, O. Torsaeter, and J.O. Aasen, eds., North Sea

Oil and Gas Reservoirs-II. Graham & Trotman, p.354-362.

Kittridge, M.G., L.W. Lake, F.J. Lucia, and G.E. Fogg, 1990. Outcrop/subsurface

comparisons of heterogeneity in the San Andreas Formation. SPE Formation Evaluation,

September 1990, p. 233-240.

KoHermann, C.E., and S.M. Gorelick, 1996. Heterogeneity in sedimentary deposits: A


Resources Research, vol 32, no. 9, p. 2617-2658.

Krumbein, W.C., 1967. Fortran IV Computer Programs for Markov Chain Experiments




164

Lantuejoul, C , 1994. Nonconditional simulation of stationary isotropic multigaussian

random functions. In: M. Armstrong and P.A. Dowd, eds., Geostatistical Simulations.

Kluwer Academic Publishers, Boston, p. 147-177.

Lin, C. and Harbaugh, J.W., 1984. Graphic Display of Two- and Three-Dimensional

Markov Computer Models in Geology. Van Nostrand Reinhold, New York.

Luo, J., 1996. Transition probability approach to statistical analysis of spatial qualitative

variables in geology. In: Forster,A., and D.F. Merriam, eds., Geologic Modeling and

Mapping. Plenum Press, New York, p.281-297.

MacDonald, A.C., and J.O. Aasen, 1994. A prototype procedure for stochastic modeling

of facies tract distribution in shoreface reservoirs. In: Yarus, J.M., and R.L. Chambers,

eds. Stochastic Modeling and Geostatistics - Principles, Methods, and Case Studies.

American Association of Petroleum Geologists Computer Applications in Geology, no.3.

AAPG, Tulsa, Oklahoma, p. 91-108.

MacMillan, J.R., and A.L. Gutjahr, 1986. Geological controls on spatial variability for

one-dimensional arrays of porosity and permeability normal to layering. In: Lake, L.W.,

and H.B. Carroll, Jr., eds., Reservoir Characterization. Academic Press, p. 265-292.

McDonald, M.G., and A.W. Harbaugh, 1988. MODFLOW: A Modular

Three-Dimensional Finite Difference Ground-Water Flow Model. Techniques of

Water-Resources Investigations of the United States Geological Survey, U.S. Geological

Survey, U.S. Department of the Interior.

Miall, A.D., 1980. Cyclicity and the facies model concept in fluvial deposits. Bulletin of

Canadian Petroleum Geology, vol. 28, no.l, p.59-80.

165

Miall, A.D., 1985. Architectural-element analysis; a new method of facies analysis applied

to fluvial deposits. Earth Science Reviews, Vol 22, no.4, p. 261-208.

Miall, A.D., 1988. Reservoir heterogeneities in fluvial sandstones: lessons from outcrop

studies. Bulletin of the American Association of Petroleum Geologists, vol. 72, no. 6, p.

682-697.

Middleton, G.V., 1973. Johannes Walther's Law of the Correlation of Facies. Geological

Society of America Bulletin, vol. 84, p. 979-988.

Moss, B.P., 1990. Stochastic reservoir description: a methodology. In: Morton, A.C., A.

Hurst, and M.A. Lovell, eds., Geological Applications of Wireline Logs. Geological

Society Special Publication No. 48, p. 57-76.

Murray, C.J., 1994. Identification and 3-D Modeling of Petrophysical Rock Types. In:

Yarns, J.M., and R.L. Chambers, eds., Stochastic Modelling and Geostatistics - Principles-

Methods, and Case Studies. AAPG Computer Applications in Geology, No. 3, p. 55-64.

Neton. M.J., J. Dorsch, CD. Olson, and S.C. Young, 1994. Architecture and directional

scales of heterogeneity in alluvial-fan aquifers. Journal of Sedimentary Research, vol.

B64. no. 2. p. 245-257.

Oertel. G.. and E.K. Walton, 1967. Lessons from a feasibility study for computer models

of coal-bearing deltas. Sedimentology, vol. 9, pp. 157-168.

Ouenes. A., and S. Bhagavan, 1994. Application of simulated annealing and other global



166





finite-difference ground-water model. U.S.G.S. Open File Report 89-381, 81 pp.

Power, B.A., and R.G. Walker, 1996. Alio stratigraphy of the Upper Cretaceous Lea

Park-Belly River transition in central Alberta, Canada. Bulletin of Canadian Petroleum

Geology, vol. 44, no. 1, p. 14-38.

Powers, D.W., and R.G. Easterling, 1982. Improved methodology for using embedded

Markov chains to describe cyclical sediments. Journal of Sedimentary Petrology, vol. 52,

no. 3, p. 913-923.

Richman, D., and W.E. Sharp, 1990. A method for detennining the reversibility of a

Markov sequence. Mathematical Geology vol. 22, no. 7, p. 749-761.



Rust, B.R., 1977. Mass flow deposits in a Quaternary sucession near Ottawa, Canada;

diagnostic criteria for subaqueous outwash. Canadian Journal of Earth Science, vol. 14, p.

175-184.

Rust, B.R., and R. Romanelli, 1975. Late Quaternary subaqeuous outwash deposits near

Ottawa, Canada. In: Jopling, A.V., and B.C. McDonald, eds., Glaciofluvial and

Glacioulacustrine Sedimentation. SEPM Special Publication No. 23. Society of

Economic Paleontologists and Mineralogists, Tulsa, Oklahoma, p. 177-192.

167

Schwarzacher, W., 1969. The use of Markov chains in the study of sedimentary cycles.

Mathematical Geology vol. 1, no. 1, p 17-39.




Geology vol.12, no. 3, p. 213-234.

Schwarzacher, W., 1993. Cyclostratigraphy and the Milankovitch Cycle. Developments

in Sedimentology 52, Elsevier, 225 pp.

Sen, M.K., A. Datta-Gupta, P.L. Stoffa, L.W. Lake, and G.A. Pope, 1995. Stochastic

reservoir modeling using simulated annealing and genetic algorithm. SPE Formation

Evaluation vol. 10, no. 1, p. 49-55.

Shanley, K.W., and P.J. McCabe, 1994. Perspectives on the sequence stratigraphy of

continental strata. American Association of Petroleum Geologist Bulletin, vol. 78, no. 4,

p. 544-568.

Sharp, W.E., 1982a. Stochastic simulation of semivariograms. Mathematical geology, vol.

14, no. 5, p .445-456.

Sharp, W.E., 1982b. Estimation of semivariograms by the maximum entropy method.

Mathematical geology, vol. 14, no. 5, p .457-474.

Silliman and Wright, 1988. Stochastic analysis of paths of high hydraulic conductivity in

porous media. Water Resources Research, vol. 24, no. 11, p. 1901-1910.

168

Smith, L., and R.A. Freeze, 1979. Stochastic analysis of steady-state groundwater flow in

a bounded domain, 2, Two-dimensional simulations. Water Resources Research, vol. 15,

no. 6, p. 1543-1559.

Smith, L.S., and F.W. Schwartz, 1980. Mass transport, 1. A stochastic analysis of

macroscopic dispersion. Water Resources Research vol. 16, no. 2. p.303-313.

Srivistava, R.M., and H.M. Parker, 1989. Robust measures of spatial continuity. In: M.

Armstrong, ed., Geostatistics .Vol. I. Kluwer Academic Publishers, p. 295-308.

Stauffer, D., 1985. Introduction to Percolation Theory. Taylor and Francis, London., 124

pp.

Turk, G., 1979. Transition analysis of structural sequences: Discussion. Geological

Society of America Bulletin, Part I, vol. 90, p. 989-991.

Vail, P.R., F. Audemard, S.A. Bowman, P.N. Eisner, and C. Perez-Cruz, 1991. The

stratigraphic signatures of tectonics, eustacy and sedimentology - an overview. In Einsele,

G., W. Ricken, and A.Sielacher, eds. Cycles and Events in Stratigraphy. Springer-Verlag,

p. 617-659.


ed. Facies Models, 1st Edition. Geoscience Canada Reprints Series 1. p. 1-8.

Weber, K.J., and L.C. van Guens, 1990. Framework for constructing clastic reservoir

simulations models. Journal of Petroleum Technology, October 1990, p. 1248-1297.

Wheeler, H.A, 1958. Time stratigraphy. AAPG Bulletin, vol. 42, no. 5, p. 1047-1063.

169

Wilgus, C , S. Hastings, C. Ross, H. Posamentier, J. Van Wagoner, and C.G. Kendall,

1988. Sea-Level Changes: an Integrated Approach. Society of Economic Mineralogists

and Paleontologists Special Publication No. 42.

Yarns, J.M., and R.L. Chambers, 1994. eds. Stochastic Modeling and Geostatistics,

AAPG Computer Applications in Geology No. 3, American Association of Petroleum

Geologists, Tulsa, Okalhoma, 379 pp.

Zeller, E.J., 1964. Cycles and Psychology. In: D.F. Merriam, ed. Symposium on Cyclic

Sedimentation. Geological Survey of Kansas Bulletin 169, Volume II. p. 631-636.

170

Appendix A

A Fortran Code for Percolation Experiments

program perc

c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

c A program to identify the maximum value of lattice-node weights, say c grid-block K, for which there is at least one connected paths between c two opposite sides of a 2-D field.

c The algorithm starts at the second row from the last. Each node is c visited in turn. All of the possible paths from that node to the last c row are identified. In an nxn grid there will be n paths to consider c for each node, nA2 paths to consider for each row, and nA3 operations to c consider for the entire grid.

c The minimum nodal weight along each path is considered. Once all c paths are considered, the node is assigned the value of the maximum c value of the set of minimum path values from that node. This value c is stored in a new array. Each node in a row is so treated, using the c original values of the row nodes each time. Once the row is complete, c the process is repeated. But this time, the end nodes of each path on c the last row will be the new assigned values, not the original values, c The row nodes remain as their original values.

c This algorithm propogates the maximum connected path value from each node c backwards on the grid. The maximum reassigned value along the bottom c row (if we started at the top of the grid) will be the extreme path c value. A subroutine will convert this value as a pdf to a cdf using c the Hastings approximation. Both values will be written a file. c

Real old(50,50), new(50,50), pathmin(50), nodemax Integer ii,i,j,jj,k,f,m double precision pi, c(6), t, u realn

open(3,file=*50by50.grid') open(4,file='perc.out') open(5,file-old.out')

c Read in the original grid into old(50,50).

Do5,j=l,50

Do 6, i=l,50

read(3,*) x,y,z,old(i,j)

6 continue

5 continue

c Read in top row (j~50) into new array

Do 15, i=l,50

new(i,50)= old(i,50)

15 continue

c Main Loop over all the rows

Do25,j=49,l,-l

c For each node

Do 35, i=l,50

Do 36, ii=l,50

pathmin(ii)=old(i,j)

36 continue

c Find the lowest nodal value for all the 50 paths to the next row c by reading the min value along each path in turn into the array c pathmin(50), checking each value.

c After all paths have been checked, the array pathmin is sorted and its c highest value, nodemax, is placed into array as new(ij)

c First do the straight ahead path.

if(new(i,j+l).lt.pathrnin(i)) then

pathmin(i)=ne w(ij+1)

endif

Now check all the right hand paths

m=i

Do44f=i+l,50

m=m+l

Do 45 k=i+l,f

if(old(kj).lt.pathmin(m)) then

pathmin(m)=old(kj)

endif

Continue

if(new(k,j+l).lt.pathmin(m)) then

pathmin(m)==new(k,j+1)

endif

Continue

Now check all the left hand paths

m=i

Do54f=i-l,l,-l

m=m-l

Do 55 k=f,i

if(old(kj).lt.pathmin(m)) then

pathmin(m)=old(k,j) endif

Continue

if(new(fj+l).lt.pathmin(m)) then

pathmin(m)=new(fj+1)

endif

54 Continue

c now sort pathmin to find the maximum value and assign that to c new(ij).

nodemax=pathmin( 1)

Do 65,jj=2,50

if (pathmin(ij).gt.nodemax) then

nodemax=pathmin(ii)

endif

65 Continue

new(i,j)=nodemax

35 Continue

25 CONTINUE

c Write old and new matrices to files.

Do75,j=50,l,-1

write(4,101) (new(ij), i=l,50) write(4,*)'' write(5,101) (old(iij), ii=l,50) write(5,*)''

75 continue

101 format(50(f5.2))

close(3)

175 close(4)

c Identify the maximum value in row 1. Calculate the cdf using c the hastings equation. Write both the Kmax and EPV to the screen.

rowmax=new( 1,1)

Do 95, ij=2,50

if(new(ij,l).gt.rowmax) then

rowmax=new(ij, 1)

endif

95 continue

c A program to generate an approximation of value of G (normal cdf) c given a normal-score value of z (Hastings algorithm).

Subroutine removed for copywrite.

Appendix B

Fortran Codes for Annealing Markov Fields

177 program manneal

c * This program generates 2 or 3-dimensional Markov fields by * c * true simulated annealing. The fields can be conditioned though * c * this degrades the convergence of the objective function. This version of c * manneal has not been validated for conditioning . c * c * The control statistics are two and three-point histograms * c * for single and double dependent Markov chains. In this version of the program, * c * the subroutine to enforce double dependency is disabled because it has not been * c * validated for the newer implementation of the perturbation method. * c * The histogram file is generated by the program Prephist.for. Only a single c * lag is enforced. Unlike empirical multipoint histograms, the * c * use of the single lag is sufficient to enforce all derived * c * probability relationships at higher lags. The input parameter * c * file is called markov.par *

c * A second parameter file is required. This file, annealpar, * c * holds the parameters for controlling the annealing routine. * c * * c * Ifconditioning data are used, they are to be found in a third * c * input file called cond.dat. The file is in GSLIB standard format, * c * with four columns: xloc, yloc, zloc in grid coordinates plus * c * data value. * c * * c * The maximum number of states is set at 6. * c * The largest grid is 500*500* 1 blocks in this version. The third dimension * c * has been effectively disabled because this option has not been validated * c * for the perturbation options now employed. * c * The important variables are as follows: * c * * c * iord: 1 or 2, the dependency of the Markov structure. This * c * determines whether two or three point histograms are enforced. * c * isx,isy,isz= grid dimensions in x,y,z c * itau: can be 1 to 5. In a double dependent structure, itau is c * lag separating the state itau steps prior from that one step c * prior to any position in a chain. c * im = number of states. * c * train,hist,sthist etc: the various arrays holding the * c * the multipoint histograms for nonconditioned points. * c * traincon,histcon,sthistcon, etc: the various arrays for holding * c * the multipoint histograms for conditioning points. * c * wn: the weighting given to conditioning data. wn=l means no *

*

*

c * additional weight given to conditioning data points. * c * temp,object,lambda,del, etc.: control parameters for annealing. *

c * This is an experimental program. Use at your own risk. No * c * warranty as to the validity of this program or its results is * c * made or implied. * c * * c * Copywrite 1997 Kevin P. Parks. All rights reserved. *

integer iord,itau,isx,isy,im,dimflag,isz,swapflg integer condflag,iflag,ktry,kaccept,maccept,mtry,maxtry integer istate,kstate,jstate,ix,iy,iz,iold(2),inew(2),ndir,imk integer iseed,storeflag,report,reports,nsims,screen integer train(6,6,6,2,3),swx(2),swy(2),swz(2) integer hist(6,6,6,2,3),histcon(6,6,6,2,3) integer sthist(6,6,6,2,3),sthistcon(6,6,6,2,3) integer grid(500,500,1 ),count,maxtrials,methflg,initflg logical condfl,accept,tdim,cond(500,500,1) realpropstate(6),wn,object,objectnew real cumprop(6),objectold,tempold realtemp,lambda,del,converge,objectstore real wl ,w2,w3,obj 1 ,obj2,obj3 integer freqx(5,5),freqy(5,5),same

c STEP 1 c Import the annealing parameters. c Also set up an automatic log file called 'history.out'. c This file is helpful for debugging. Open the output files.

open(8 ,file='history. out')

open(l Ufile^'anneaLpar')

c skip the header read(ll,*)

c read the initial temperature, usually this is one. read( 11,111) tempo Id print *, 'temp =', tempold

c Read in lambda, the muliplier on temperature. read(l 1,111) lambda

print *, 'lambda = ',lambda

c Read in the number of acceptance perturbations which reduce c the objective function to be required at any given c temperature before cooling. This is usually 10*number of c grid blocks but can be fewer.

read( 11,112) maccept print *, 'maccept =' , maccept

c Read in the maximum number of perturbations to be tried at c any given temperature before cooling. Usually this is c 100* number of grid blocks but can be fewer.

read(ll,112)mtry print *, mtry = ', mtry

c Read in the maximum number of attempts at lowering temperature c after mtry exceeds maccept. Usually lowering temperature once c mtry exceeds maccept is futile, so set maxtrials to be 0 or 1.

read(l 1,112) maxtrials print *, 'maxtrials = ',maxtrials

c Read in the convergence criterion. This is the value of objective c function (normalized to the initial value) that must be reached in c order to consider the annealing to be completed. Values of 0.00001 c or 0.000001 are good initial targets. Some objective functions will c not be able to reach such low values because they are somehow c internally inconsistent. Conditioned Markov fields are affected by c such a pathology. Simulated annealing's appeal is that it can still c find a local minimum is such pathological objective functions despite c such internal inconsistencies.

read( 11,111) converge print *,'converge = ',converge

c Read the reporting schedule to the screen and debug file.

read(l 1,112) reports print *,'reporting interval to screen is', reports print *,'reporting interval to debug file is', reports screen=reports

Read the number of simulations (realizations) to produce.

read(ll,112)nsims print *, "Number of Simulations', nsims

format(£20.6) format(i20)

read( l l , l l l )wl print *,'off-diagonal weight :',wl read(l l , l l l )w2 print *,'diagonal weight :',w2 read(l l , l l l )w3 print *,'weight on global histogram penalty :',w3 read(ll,112)swap£lg print *,'swapflg = ',swapflg read(ll,112)methflg print *,'method flag-, methflg read(ll,112)initflg print *,'initflag=',initflg read the random number seed read(ll,112) iseed close(ll)

In addition to the debug file, two files are also openned. The file out.file contains the resulting layer by layer categorical arrays followed by the final multipoint histogram arrays. The file gslib.dat is a GSLIB formatted file of the same data.

open(l 6,file='out.file') open( 17,file-gslib.dat') write(17,*)'Test File' write(17,*)T write( 17,*) 'State' do 1000, isim=l,nsims

temp=tempold

STEP 2 Initialize all the arrays and logical flags

caUimtial(iord,itau,isx,isy,isz,im,dimflag,condflag, iflag,ktry, kaccept,iold,inew,

* storeflag,train, hist,histcon,sthist,sthistcon,grid,condfl, * accept,tdim,cond,propstate,wn,object,objectnew, * cumprop,objectold,objrsc)

c STEP 3 c Read in the multipoint histograms and grid parameters from c the parameter file 'markov.parm'

callreadparm(iord,itau,isx,isy,tdim,isz, * train,condfl,propstate,im,wn,ndir,imk)

c STEP 4 c Generate the Initial Image in 'grid' and set conditioning flags.

call init(im,isx,isy,isz,propstate, * condfl,grid,cond,cumprop,initflg, * tdim,iseed)

c TEST OUTPUT FROM INIT: WRITE OUT GRID AND COND ARRAY do 15, k=l,isz write(8,*) 'Initial Grid for Layer: ',k do 15,j=isy,l,-l

write(8,103) (grid(i,j,k),i=l,isx) 103 format(500i3) 15 continue

c Write out conditioning array for debugging.

c do 16, k=l,isz c do 16, j=isy,l,-l c write(8,*) (cond(i,j,k),i=l,isx) cl6 continue

c STEP 5 c Calculate the initial image's histograms.

if(iord.eq.l) then calltwphist(hist,histcon,grid,isx,isy,isz,cond,

* tdim) endif

if(iord.eq.2) then

callthrphist(hist,histcon,gnd,isx,isy,isz,cond, * tdim,itau)

endif

c TEST OUTPUT FROM TWPHIST OR THRPfflST: WRITE OUT fflSTOGRAM c ARRAY

write(8,*) 'iord = ',iord do 20, in=l,2 do 20, istate=l,im do 20,jstate=l,im do 20, kstate=l,imk do 20, idir=l,ndir

write(8,*) istate,jstate,kstate,in,idir, * hist(istate,jstate,kstate,in,idir) * histcon(istate,jstate,kstate,in,idir)

20 continue

c STEP 6 c Initialize the objective function(s) by calculating c the squared differences between the histograms of the c desired field, held array 'train', and the field being c generated, held in arrays hist and histcon.

caUobjcalc(hist,histcon,train,object,im, * isx,isy,isz,wn,tdim,imk,count,report, * wl,w2,w3,objl,obj2,obj3)

c Rescale the initial objective function to 1.0000. All objective c functions calculated will be rescaled by the same factor (objrsc). c Store the value as objectold.

objrsc=l/object objectold=objrsc*object

c write(8,*) 'object, objectold = ', object, objectold

count = 0

C Enter the main loop to anneal the image in 'grid'.

14 count=count+l

c STEP 7 c Test to see if the objective function value, rescaled by the

c original value, has decreased to a very small number held as c the variable 'converge', which was read from 'annealpar'.

c write(8,*) 'objectold,converge,count',objectold,converge,count

if(objectold.gt.converge) then

c Write progress to screen (and log file) as needed

if((int(count/screen)*screen).eq.count) then c print *, 'objectold,temp,count,kaccept,ktry,rnaxtry,nsims'

print *, objectold,temp,count,kaccept,ktry,isim c write(8,*),'count,temp,ktry,kaccept,objectold,der, c * count,temp,ktry,kaccept,objectold,del

endif

c STEP 8 c Peturb the field by repicking a nonconitioning node at random c from the global proportion (S WAPFLG^l) or by swapping two nonconditioning c nodes picked at random (SWAPFLG=2)

c Perturb a nonconditioning node at random.

IF(SWAPFLG.EQ.l) THEN

4 call nodepick(isx,isy,isz,ix,iy,iz,tdim,iseed)

C Check that selected node is not a conditioning node. If not, C then store old state and pick a new state, ensuring that C the new state is not the same as the old state

if (.not.cond(ix,iy,iz)) then

iold( 1 )=grid(ix,iy,iz) swx(l)=ix swy(l)=iy swz(l)=iz

call newstate(im,cumprop,inew,iseed)

if(inew(l).eq.iold(l)) then go to 4

endif else

go to 4 endif

ENDIF

IF(SWAPFLG.EQ.2) THEN

51 call nodepick(isx,isy,isz,ix,iy,iz,tdirn,iseed)

if(.not.cond(ix,iy,iz) then iold( 1 )=grid(ix,iy,iz)

endif

c Store the coordinates swx(l)=ix swy(l)=iy swz(l)=iz

callnodepick(isx,isy,isz,ix,iy,iz,tdim,iseed)

if(.not.cond(ix,iy,iz)) then iold(2):=grid(ix,iy,iz)

endif

if(iold(l).eq.iold(2)) then goto 51

endif

c Store the coordinates swx(2)=ix swy(2)=iy swz(2)=iz

C Do the swap

INEW(l)=IOLD(2) INEW(2)=IOLD(l)

ENDIF

c STEP 9 c Store the Old Histograms in case perturbation is

storeflag=0

* call storeold(hist,histcon,im, tdim,stWst,sthistcon,storeflag,imk)

c STEP 10 c Store Old Object in case perturbation is rejected

objectstore=object

c STEP 11 c Update the histograms

if(iord.eq.l) then call update(isx,isy,isz,iold,inew,

* tdim,swx,swy,swz,cond,grid,hist,histcon,im,swapflg, * count,initflg)

endif

if(iord.eq.2) then call thrupdate(isx,isy,isz,iold,inew,

* tdim,ix,iy,iz,cond,grid,hist,histcon,itau) endif

c STEP 12 c Recalculate the Objective Function with Perturbation

call objcalc(hist,histcon,train,object,im, * isx,isy,isz,wn,tdim,irnk,count,report,

wl,w2,w3,objl,obj2,obj3) *

objectnew=object*objrsc

c STEP 13 c Determine if perturbation is accepted. It will be accepted if c it reduces the value of the objective function. If it increases c the value of the objective function, the perturbation will be c accepted with a probability P=exp(-del/T), where del=objectnew c minus objectold rescaled by objrscl, the initial value. c If pertubation is rejected, reset histograms, object values.

del=objectnew-objectold

accept=.false.

if(del.lt.O) then accept=.true.

IF(SWAPFLG.EQ.l) THEN ix=swx(l) iy=swy(l) iz=swz(l) grid(ix,iy,iz)=inew( 1) kaccept = kaccept+1 ktry=ktry+l objectold=objectnew go to 11

ENDIF

IF(SWAPFLG.EQ.2) THEN ix=swx(l) iy=swy(l) iz=swz(l) GRID(ix,iy,iz)=INEW( 1) ix=swx(2) iy=swy(2) iz=swz(2) GRID(ix,iy,iz)=INEW(2) kaccept = kaccept+1 ktry=ktry+l objectold=objectnew go to 11

ENDIF

endif

if(del.gt.0.and.methflg.eq.l) then call boltzman(deLtemp,iseed,accept)

endif

if^methflg.eq. 1 .and.swapflg.eq. 1 .and.accept) then kaccept=kaccept+1 ktry=ktry+l objectold=objectnew ix=swx(l) iy=swy(l)

iz=swz(l) grid(ix,iy,iz)=inew( 1) go to 11

endif

if(methflg.eq. 1 .and.swapflg.eq.2.and.accept) then kaccept=kaccept+1 ktry=ktry+l objectold=objectnew ix=swx(l) iy=swy(l) iz=swz(l) grid(ix,iy,iz)=inew( 1) ix=swx(2) iy=swy(2) iz=swz(2) grid(ix,iy,iz)=inew(2) go to 11

endif

if(.not.accept) then Reset all

storeflag=l call storeold(hist,histcon,irn, tdim,sthist,sthistcon,storeflag,imk) object=objectstore ktry=ktry+l

endif

STEP 14: Determine if kaccept is less than target number acceptances for any given temperature. If so, go back to 14 and do another perturbation. Do this only if ktry = the total of all pertubrations, not just the accepted ones, is less than mtry, the total number of perturbations attempted at any temperature.

if(kaccept.lt.maccept.and.ktry.lt.mtry) then go to 14

endif

c STEP 15 c If kaccept reaches maccept before ktry reaches mtry, lower c the temperature by multiplying by the constant lambda.

if(kaccept.ge.maccept.and.ktry.lt.mtry)then temp=temp* lambda uttemp.lt.10e-15) then

print *,'Temperature below 10e-15. Ending program.' go to 777

endif

c Reset counters for new temperature ktry=0 kaccept=0 go to 14

endif

c STEP 16 c If the target number of acceptable perturbations, maccept,is c not reached before the maximum allowable perturbations are tried c at a given temperature, try reducing the temperature anyways. c This can be done up to a total of maxtrials times. There is not c usually any point to make maxtrials go past 1.

if(ktry.ge.mtry) then c change temperature

temp=temp* lambda c reset counters

kaccept=0 ktry=0 maxtry=maxtry+1

endif

c send back ifmaxtry not reached if(maxtry.lt.maxtrials) then

go to 14 endif

c STEP 17 c If temperature cannot be lowered further and the convergence c criteria is not met, print a message to the screen and send c the results to the output file.

if(maxtry.ge.maxtrials) then

uttemp.lt.10e-

write(8,*) 'maxtry,maxtrials',maxtry,maxtrials print *, 'Maxtrials Reached, Check Annealing Schedule' print *, 'Cannot cool any further.'

8788 print *, 'Kicking out of Program' print *, 'Count is ',count print *, 'Kaccept,Ktry,Maxtry' print *, kaccept,ktry,maxtry

endif

c If the objective function has been reduced to the convergence c you will end up here. Write the output.

endif

777 do 25, ik=l,isz write(16,*) 'Layer', ik

do 25, i=isy,l,-l write(16,104) (grid(j,i,ik)j=l,isx)

104 format(200(i3)) 25 continue

c Write out the histograms write(8,*) write(8,*) 'Global Histogram' do 732, i=l,im do 732, k=l,imk

write(8,*) i,i,k,idir,train(Li,k, 1,1), * hist(i,i,k, 1,1 ),histcon(i,i,k, 1,1)

732 continue write(8,*) write(8,*) "Updated Histograms' do 733, i=l,im do 733,j=l,im do 733, k=l,imk do 733, idir=l,ndir

write(8,*) i,j,k,idir,train(i,j,k,2,idir), * hist(i,j,k,2,idir),histcon(i,j,i,2,idir)

733 continue

write(8,*)

write(8,*)

wnte(8,*) write(8,*) 'Components of objective function:' write(8,*) 'Off diagonal component, weight: ',objl,wl write(8,*) 'Diagonal component,weight: ',obj2,w2

c Write output to a GSLIB formatted File c The array will be written out "upside down" because c the origin for a GSLIB file is the lower left corner of c the grid.

write(17,*) 'manneal.out' write(17,*) T write(17,*) 'Category'

do 28, i=isy,l,-l do 28,j=l,isx

write(17,*)grid(j,U) 28 continue

c do 26, i=isy,l,-l c write( 16,*) (cond(j,i, 1 ),j= 1 ,isx) c26 continue

105 format(8i8)

print *, 'Final Objective Function = ',objectold print *, 'Final Count = ',count,' iterations.'

1000 continue

close(17) close(8) close(16) close(21)

999 end

A 3JC !|C 3|C «|C «JC 3|C 3)C 3JC 5JC 3fC 3|C JfC *fC - p ^ . ^C J|C 5JC 3|C 3|C *(C *fC »JC #(C »1C «fC - f . J(C ?|C 3fC 3f» -JC #JC *jC 3fC 9yC 3|C 5fC 5fC 3JC 3JC 3|C 3|C 3|C 3|C 3JC ;fC 5JC ?|5 3|C *f> T * * 1 * * 1 * T * * P * P * F * r n * T * ^ * F * l * * l * *r ^ T *

subroutmeinitial(iord,itau,isx,isy,isz,im,dimflag,condflag, * iflag,ktry,kaccept,iold,inew, * storeflag,train, hist,histcon,sthist,sthistcon,grid,condfl, * accept,tdim,cond,propstate,wn,object,objectnew, * cumprop,objectold,objrsc)

c *******************************************************************

integer iord,itau,isx,isy,isz,im,dirnflag integer condflag,iflag,ktry,kaccept integer iold,inew,iseed,storeflag integer train(6,6,6,2,3) integer hist(6,6,6,2,3),histcon(6,6,6,2,3) integer sthist(6,6,6,2,3),sthistcon(6,6,6,2,3) integer grid(500,500,l) logical condfl,accept,tdim,cond(500,500,1) realpropstate(6),wn,object,objectnew real cumprop(6),objectold,objrsc

do 10,i=l,6 do 10, j=l,6 dol0,k=l ,6 do 10, in=l,2 do 10, idir=l,3

train(i,j,k,in,idir)=0 hist(i,j,k,in,idir)=0 histcon(i,j,k,in,idir)=0 sthist(i,j,k,in,idir)=0 stnistcon(i,j,k,in,idir)=0

10 continue

do 20, i=l,500 do20,j=l,500 do20,k=l,l

grid(i,j,k)=0 cond(i,j,k)=.false.

20 continue

do 30 i=l,6 propstate(i)=0 cumprop(i)=0

30 continue

object=0 objectnew=0 objrsc=l objectold=0 iflag=0 condflag=0 storeflag=0 ktry=0

kaccept=0 dimflag=0 condfKfalse. accept=.false. tdim=.false. object=0 wn=l iold=l inew=l isx=0 isy=0 isz=0 iord=l itau=l im=l

return end

c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

subroutine readparm(iord,itau,isx,isy,tdim,isz,

* train,condfl,propstate,im,vvn,ndir,imk)

c This subroutine reads the grid paramters plus the multipoint c histogram created by Prephist.for.

integer iord4tau,isx,isy,irn,dimflag,isz integer condflag integer istate,kstate,jstate,ndir,imk integer train(6,6,6,2,3) logical condfl,tdim real propstate(6),wn

open(3 ,file='markov.parml)

read(3,*)im

read(3, *) (propstate(i),i= 1 ,im) 102 format(10f5.2)

read(3,*)iord if(iord.gt.l) then

read(3,*) itau endif

read(3,*) isx read(3,*) isy read(3,*) dimflag tdim=.false. if(dimflag.gt.O) then

tdim=.true. read(3,*) isz

else isz=l

endif

read(3,*) condflag if(condflag.gt.O) then

condfl=.true. read(3,*) wn

else wn=1.0

endif if(tdim) then

ndir=3 else

ndir=2 endif

c Set up the training histograms such that for lag=0 (lag register 1) c the state-to-self state histogram equals the proportion of the state c time the dimension of the grid, c For 1 st order grids, consider that kstate is always 1.

if(iord.eq.l) then imk=l

else imk=im

endif

c Initialize Global Proportions tsum=0 numgrid=real(isx)*real(isy)*real(isz)

do 5, istate=l,im-l if(iord.eq.l) then

train(istate,istate, 1,1,1)=

* nint(propstate(istate)*numgrid) tsum=tsum+train(istate,istate, 1,1,1)

endif

if(iord.eq.2) then train(istate,istate,istate, 1,1)=

* int(propstate(istate)*numgrid) endif

5 continue

train(im,im, 1,1,1 )=numgrid-tsum

c Read in multipoint histograms

do 10, istate=l,im do 10, jstate=l,im do 10, kstate=l,imk do 10, idir=l,ndir

read(3,*)train(istate,jstate,kstate,2,idir) 101 format(ilO) 10 continue

close(3)

return end

Q*********************************************************************

subroutine init(im,isx,isy,isz,propstate,condfl,grid, * cond,cumprop,initflg,tdim,iseed)

p*********************************************************************

C This subroutine generates the random image to anneal with the C appropriate proportion of each state.

C Variables are: C isx,isy,isz = grid dimensions C im = total number of states C propstate(im) = stable proportion of each state C C If initflg= 1, generate a random image with random proportions C provided by a random number generator C If initflg=2, generate a random image with strict proportions of C of the markov parameter file

C If initflg=3, read in a previous image from the file "out.fil". C Q I t : * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

integer im,isx,isy,isz,iseed,val(6),invari,total integer grid(500,500,1 ),ix,iy,iz,initflg,tdim,num(6) logical cond(500,500,1 ),condfl,occ(500,500,1) real propstate(6) real cumprop(6) external ranO

c First, convert the proportion of each state to a c cumulative probability.

cumprop( 1 )=propstate( 1) cumprop(im)=l .000 do 10, i=2,im-l

cumprop(i)=cumprop(i-1 )+propstate(i) 10 continue

c There are two arrays. One is the array called 'grid' that c actually holds the simulation values. The other is called c 'cond'. The 'cond' array holds logical flags correspondent c to each location in 'grid' that tells whether it is a conditioning c point or not. The next steps initialize the 'cond' array while c generating initialize values for the array 'grid'.

if(initfig.eq.l) then

do 20, i=l,isx do 20, j=l,isy do 20, k=l,isz

cond(i,j,k)=.false. x =ran0(iseed) do 30, istate=im,l,-l

if(x.le.cumprop(istate)) then grid(i,j,k)=istate

endif 30 continue 20 continue

endif

if(initflg.eq.2) then

total=0 do 110, is=l,im-l

num(is)=nint(propstate(is) * isx* isy* isz) total=total+num(is)

110 continue num(im)=(isx* isy* isz)-total

do 120,iz=l,isz do 120, iy=l,isy do 120, ix=l,isx

occ(ix,iy,iz)=.false. 120 continue

do 130, i=l,im do 135,j=l,num(i)

15 call nodepick(isx,isy,isz,ix,iy,iz,tdim,iseed) if(.not.occ(ix,iy,iz)) then

grid(ix,iy,iz)=i occ(ix,iy,iz)=.true.

else go to 15

endif 135 continue 130 continue

endif

if(initflg.eq.3) then open(45, file='out.fiT)

do 35, iz=l,isz do 35, iy=isy,l,-l

read(45,*) (grid(ix,iy,iz),ix=l,isx) 35 continue

close(45) endif

C Add in the conditioning nodes which are in the conditioning file. C The conditioning file is in GSLIB format, with x,y,z and

197 C state as columns in that order. Coordinates must be input as grid C coordinates. The conditioning flag (condfl) is read in readparm. C Adding in conditioning data will destroy a perfectly proportioned C random field.

if(condfl) then open(l l,file='cond.dat')

read(ll,*) 102 format(i5)

read( 11,102) invari do 40 i=l,invari

read(ll,*) 40 continue

c Read all the data until the end of the file:

50 read(l l,*,end=60) (val(j)j=l,4) ix=val(l) iy=val(2) iz=val(3) grid(ix,iy,iz) = val(4) cond(ix,iy,iz) = .true, go to 50

60 close(ll) endif

c

c

return end

subroutine nodepick(isx,isy,isz,ix,iy,iz,tdim,iseed)

c This subroutine picks a node at random from a field of dimension c isx,isy,isz.

integer isx,isy,isz,ix,iy,iz,iseed realx

logical tdim external ranO

10 x=ranO(iseed) ix=nint((x)*real(isx)) if(ix.le.0.or.ix.gt.isx) then

go to 10 endif

11 x=ranO(iseed) iy=nint((x)*real(isy))

if(iy.le.0.or.iy.gt.isy) then go to 11

endif

12 if(tdim)then x=ranO(iseed)

I z=nint((x)*real(isz)) else

iz=l endif

if(iz.le.0.or.iz.gt.isz) then go to 12

endif

return end

p * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

subroutine newstate(im,cumprop,inew,iseed)

C This subroutine picks a new state at random from the C cumulative proportions of available states.

integer im,iseed,inew(2),istate real cumprop(6) external ranO

x =ranO(iseed)

do 30, istate=im,l,-l if(x.le.cumprop(istate)) then

inew(l)=istate endif

30 continue

return end

c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

subroutine storeold(hist,histcon,im, * tdim,sthist,sthistcon,storefkg,irnk)

c This subroutine stores the old histograms in case a perturbation is rejected. c **********************************************************************

integer im,storeflag,imk integer hist(6,6,6,2,3),histcon(6,6,6,2,3) integer sthist(6,6,6,2,3),sthistcon(6,6,6,2,3) logical tdim

if (storeflag.eq.O) then do 10, in=l,2 do 10, idir=l,2 do 10, istate=l,im do 10,jstate=T,im do 10, kstate=l,imk

sthist(istate,jstate,kstate,in,idir)= * hist(istate,jstate,kstate,in,idir)

sthistcon(istate,jstate,kstate,in,idir)= * histcon(istate,jstate,kstate,in,idir)

10 continue

if(tdim) then do 15, in=l,2 do 15, istate=l,im do 15, jstate=l,im do 15, kstate=l,imk

sthist(istate,jstate,kstate,in,3)= hist(istate,jstate,kstate,in,3) sthistcon(istate,jstate,kstate,in,3)= histcon(istate,jstate,kstate,in,3)

continue endif

*

*

15

endif

200 if (storeflag.eq.l) then

do 20, in=l,2 do 20, idir=l,2 do 20, istate=l,im do 20,jstate=l,im do 20, kstate=l,imk

hist(istate,jstate,kstate,in,idir)= * sthist(istate,jstate,kstate,in,idir)

histcon(istate,jstate,kstate,in,idir)= * sthistcon(istate,j state,kstate,in,idir)

20 continue

if (tdim) then do 25, in=l,2 do 25, istate=l,im do 25,jstate=l,im do 25, kstate=l,imk

hist(istate,jstate,kstate,in,3)= * sthist(istate,jstate,kstate,in,3)

histcon(istate,jstate,kstate,in,3)= * sthistcon(istate,jstate,kstate,in,3)

25 continue endif

endif

return end

„ < > * * * * * * * * # * * * * * * * * * * * * * * * # * * * # * # * * * * * * * # * # * * * * * * * * * * * * * * * * * * * * * * * * * * * *

subroutine boltzman(del,temp,iseed,accept)

c This subroutine decides whether to accept or reject a perturbation

integer iseed real temp,deLprobacc real x logical accept external ranO

probacc=exp(-del/temp)

x=ranO(iseed)

if(x.lt.probacc) then accept=.true.

else accept=.false.

endif

return end

c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

subroutine twphist(hist,histcon,grid,isx,isy,isz,cond, * tdim)

c This routine calculates the two-point histogram of an image. The c calculation is edge-wrapped to minimize boundary effects. The histogram c is stored in an array (ij,k,n,id), where i=state (1 to im) at x and j , c in the horizontal when id=l ,2 and in the vertical up and down when c id=3. Register k is for three point histograms and not used here. c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * : < = * * * * * * * * * * * * * * * * *

integer istatej state integer hist(6,6,6,2,3),histcon(6,6,6,2,3) integer isx,isy,isz integer grid(500,500,l) logical tdim, cond(500,500,l)

c reinitialize the hist arrays

do 10,i=l,6 dol0, j=l ,6 do 10, k=l,6 do 10, in=l,2 do 10, idir=l,3

hist(Lj,k,in,idir)=0 10 continue

c Calculate histograms in ndir=l :positive x c While we're at it, calculate the global proportions.

do 100,iz=l,isz do 100,iy=l,isy do 100,ix=T,isx

istate=grid(ix,iy,iz) hist(istate,istate, 1,1,1 )=hist(istate,istate, 1,1,1)+1

100 continue

do 200, iz=l,isz do 200, iy=l,isy do 200, ix=l,isx

istate=grid(ix,iy,iz) c Edge wrap the calculation if needed

if(ix+l.gt.isx) then ixw=ix+l-isx

else ixw=ix+l

endif

jstate=grid(ixw,iy,iz)

if(cond(ix,iy,iz).or.cond(ixw,iy, 1)) then histcon(istate,jstate, 1,2,1 )=

* histcon(istate,jstate, 1,2,1)+1 else

hist(istate,jstate, 1,2,1 )= * hist(istate,jstate,l,2,l)+l

endif

200 continue

Calculate histograms in ndir=2:positive y

do 400, iz=l,isz do 400, ix=l,isx do 400, iy=l,isy

istate=grid(ix,iy,iz)

Edge wrap the calculation if needed if(iy+l -gt.isy) then

iyw=iy+l-isy else

iyw=iy+l endif

jstate=grid(ix,iyw,iz)

if(cond(ix,iy,iz).or.cond(ix,iyw,iz))then histcon(istate,jstate, 1,2,2)=

* histcon(istate,jstate, 1,2,2)+1 else

hist(istate,jstate, 1,2,2)= * hist(istate,jstate, 1,2,2)+1

endif

400 continue

if(tdim) then

c Update histograms in ndir=3:positive z


istate=grid(ix,iy,iz)

c Edge wrap the calculation if necessary

if((iz+l).gt.isz) then izw=iz+l-isz

else izw=iz+l

endif

jstate=grid(ix,iy,izw)

if(cond(ix,iy,iz).or.cond(ix,iy,izw)) then histcon(istate,jstate, 1,2,3)=

* histcon(istatejstate,l,2,3)+l else

hist(istate,jstate, 1,2,3)= * hist(istate,jstate,l,2,3)+l

endif 600 continue

endif return end

204

subroutine objcalc(hist,Wstcon,train,object,im, * isx,isy,isz,wn,tdim,imk,count,report, * wl,w2,w3,objl,obj2,obj3)

j-\ * p j p 9p 3 p 3fl* 3|C *f» 5JC 3|t J p ^C * p Jj ; 3JC 3|C 5|C 7(C 3JC J j * * p ?JC JJC 3JC 3JC 3JC 5|C 3JC 3 ^ * f 3 ^ 3|C 3|C 3JC 3 p 3 p 3f* JJC » f J p 3jC JJ* J|C *(t JfC JJC 3 f ^ - J ^ 3JC 3f* *fC 3 ^ *>S 3p * F * F * F * F * F * 1 * •** f ' T * * F * ^ ^ * I * * 1 * *!* *t*

c This subroutine calculates the objective function for a 2 or 3-point histogram c array hist[con](istate,jstate,kstate,2,idir). The number of state-to-state c transitions in each direction are scaled by the number of total transitions in c each direction. For edge wrapped fields, the rescaling factor is c isx and isy.

c Conditioning pairs are weighted by wn>l whereas nonconditioning pairs are c given a weighting of 1.0 when calculating the objective function.

c The objective function is calculated globally. c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

real wn,object,x,wl,w2,objl,obj2,w3,obj3 integer isx,isy,isz,imk,ndir,count,report integer hist(6,6,6,2,3),histcon(6,6,6,2,3) integer train(6,6,6,2,3),im,istate,jstate,kstate real vhist,vhistcon,vtrain,rsumt(6,3),rsumh(6,3) logical tdim

object = 0.0 obj 1=0.0 obj2=0.0 obj3=0.0

if(tdim) then ndir=3

else ndir=2

endif

c Put the histogram arrays in a dummy variable for shorthand c The global histograms will not be enforced in the objective function. c Calculate the component objective function of the offdiagonals c (related to the embedded matrix).

do 10, idir=l,ndir do 10, istate=l,im do 10, jstate=T,im

do 10, kstate=l,imk

vhist=real(hist(istate,jstate,kstate,2,idir)) vhistcon=real(histcon(istate,jstate,kstate,2,idir)) vtrain:=real(train(istate,jstate,kstate,2,idir))

c In this version, the objective function is split into two component c parts: one the embedded Markov Chain,the other the total Markov Chain. c The weights are found independently. NOTE: This version cannot condition c data.

c The weighting of conditioning c data is flexible. A weighting of 10 to 15 is recommended. The formula is: c obj = Sum of (Ptrain-Pobs)A2 or Sum of (Ptrain-Pnon)A2 and wn(Ptrain-Pcon)A2. c To convert the raw transtion tallies to weighted probabilities, the raw sums are c divided by the nonweighted numbers of occurrences.

IF(istate.ne.jstate.and.vtrain.ne.O) THEN

x=((vtrain-vhist) * * 2)/vtrain

ENDIF

IF(istate.ne.jstate.and.vtrain.eq.O)THEN x=((vtrain-vhist)**2)

ENDIF

objl=objl+x 10 continue

c calculate the component objective function related to the c diagonals of the Markov matrix c write(21,*) c write(21, *) 'Diagonals'

do 20, idir=l,ndir do 20, istate=T,im do 20,jstate=l,im do 20, kstate=l,imk

vhist=real(hist(istate,jstate,kstate,2,idir)) vhistcon=real(histcon(istate,jstate,kstate,2,idir)) vtrain=real(train(istate,jstate,kstate,2,idir))

IF(istate.eq.jstate.and.vtrain.ne.O) THEN

x=(( vtrain- vhist) * * 2)/vtrain

ENDIF

obj2=obj2+x

20 continue

c Add a penalty term to limit departures from the global histogram.

do 30, istate=l,im vhist=real(hist(istate,istate, 1,1,1)) vhistcon=real(histcon(istate,istate, 1,1,1)) vtrain=real(train(istate,istate, 1,1,1)) x=( vtrain- vhist) * * 2/vtrain obj3=obj3+x

30 continue

object=wl *obj 1 +w2*obj2+w3 *obj3

c Comment out this statement after weights are determined

c write(21,*) object,objl,obj2,obj3 c write(21,*)

return end

subroutine update(isx,isy,isz,iold,inew, * tdimswx,swy,swz,cond,grid,hist,histcon,im,swapflg,count, * initflg)

c This subroutine updates the objective function after a perturbation. ., * * * * * * * * * * * * * * * : ! : * * * * * * * * * * * * * * * * * * * * : ( * * * * * * * * * * * * * * * * * * * * * * * * * *

integer isx,isy,isz,iold(2),inew(2),swapflg,COUNT integer bc,iy,iz,rxw,iyw,izw,swx(2),swy(2),swz(2)

logical tdim, cond(500,500,l) integer hist(6,6,6,2,3),histcon(6,6,6,2,3) integer grid(500,500,1 ),inswp,nswp,xnb,ynb,im integer initflg

IF(SWAPFLG.EQ.l) THEN NSWP=1

ENDIF

IF(SWAPFLG.EQ.2) THEN NSWP=2 xnb=0 ynb=0

ENDIF

If(swapflg.eq.2) THEN

c Check to see if swap pair are neighbours

if(swx( 1 ).eq.swx(2).and.abs(swy( 1 )-swy(2)).eq. 1) then ynb=l

endif

if(swy( 1 ).eq.swy(2).and.abs(swx( 1 )-swx(2)).eq. 1) then xnb=l

endif

if(swx(l).eq.swx(2).and.swy(l).eq.l.and.swy(2).eq.isy) then ynb=l

endif

if(swx( 1 ).eq.swx(2).and.swy( 1 ).eq.isy.and.swy(2).eq. 1) then ynb=T

endif

if(swy(l).eq.swy(2).and.swx(l).eq.l.and.swx(2).eq.isx) then xnb=l

endif

if(swy(l).eq.swy(2).and.swx(l).eq.isx.and.swx(2).eq.l) then xnb=l

endif

if(xnb.eq.l.or.ynb.eq.l) then

c Make the swap in grid,call twphist to update the c histograms by brute force. Then undo the swap and c go to the end of this subroutine

ix=swx(l) iy=swy(l) iz=swz(l) grid(ix,iy,iz)=inew( 1)

ix=swx(2) iy=swy(2) iz=swz(2) grid(ix,iy,iz)=:inew(2)

caUtwphist(Wst,histcon,grid,isx,isy,isz,cond, tdim)

Put the grid back the way it was.

ix=swx(l) iy=swy(l) iz=swz(l) grid(ix,iy,iz)=iold( 1)

ix=swx(2) iy=swy(2) iz=swz(2) grid(ix,iy,iz)=iold(2)

goto 888 endif

endif

DO 800, INSWP=1, NSWP

ix=swx(inswp) iy=swy(inswp) iz=swz(inswp)

Update the global counts - nonconditioning nodes only

hist(IOLD(INS WP),IOLD(IN S WP), 1,1,1)= hist(IOLD(INS WP),IOLD(INS WP), 1,1,1)-1

hist(INEW(INS WP),INEW(INS WP), 1,1,1)= hist(INEW(INSWP),INEW(INS WP), 1,1,1 )+l

Subtract contribution of iold for all lags in positive x. This must be done for the case of i(x)=iold and the case when i(x-l)=iold

if((ix+l).gt.isx) then ixw=ix+l-isx

else ixw=ix+l

endif


if(cond(ixw,iy,iz)) then histcon(IOLD(INS WP) jstate, 1,2,1 )= histcon(IOLD(INS WP) jstate, 1,2,1)-1

else

hist(IOLD(rNS WP) jstate, 1,2,1 )= hist(IOLD(INS WP) jstate, 1,2,1)-1

endif

if((ix-l).lt.l)then ixw=ix+isx-l

else ixw=ix-l

endif


if(cond(ixw,iy,iz)) then histcon(jstate,IOLD(INS WP), 1,2,1 )= histcon(jstate,IOLD(INSWP), 1,2,1)-1

else

hist(jstate,IOLD(INS WP), 1,2,1 )= hist(jstate,IOLD(INS WP), 1,2,1)-1

endif

Add in contribution of inew for all lags in positive x

if((ix+l).gt.isx) then

210 ixw=ix+l-isx

else ixw=ix+l

endif


if(cond(ixw,iy,iz)) then histcon(INE W(INS WP),jstate, 1,2,1 )= histcon(INE W(INS WP),jstate, 1,2,1)+1

else

hist(INE W(INS WP),jstate, 1,2,1 )= hist(INE W(INS WP),jstate, 1,2,1)+1

endif

if((ix-l).lt.l)then ixw=ix+isx-l

else ixw=ix-l

endif


if(cond(ixw,iy,iz)) then

histcon(jstate,rNEW(rNS WP), 1,2,1)= histconGstate,INEW(INS WP), 1,2,1 )+l

else

hist(jstate,rNE W(INS WP), 1,2,1)-hist(jstate,INEW(INS WP), 1,2,1)+1

endif

C Subtract out contribution of IOLD(INSWP) for all lags in positive y

if((iy+l).gt.isy) then iyw=iy+l-isy

else iyw=iy+l

endif


if(cond(ix,iyw,iz)) then histcon(IOLD(INSWP),jstate, 1,2,2)= histcon(IOLD(INS WP),jstate, 1,2,2)-1

else

hist(IOLD(INS WP),jstate, 1,2,2)= hist(IOLD(INS WP),jstate, 1,2,2)-1

endif

if((iy-l).lt.l)then iyw=iy-l+isy

else iyw=iy-l

endif


if(cond(ix,iyw,iz)) then histcon(jstate,IOLD(INSWP), 1,2,2)= histcon(jstate,IOLD(INS WP), 1,2,2)-1

else

histOstate,IOLD(INS WP), 1,2,2)= histGstate,IOLD(INSWP), 1,2,2)-1

endif

Add in contribution of INEW(INSWP) for all lags in positive y


else iyw=iy+l

endif

212


if(cond(ix,iyw,iz)) then histcon(INEW(INSWP),jstate, 1,2,2)=

* histcon(INEW(INSWP),jstate,l,2,2)+l

else

hist(INEW(INSWP)Jstate, 1,2,2)= * hist(INEW(INSWP),jstate,l,2,2)+l

endif

if((iy-l).lt.l)then iyw=iy-l+isy

else iyw=iy-l

endif


if (cond(ix,iyw,iz)) then

histcon(jstate,rNEW(INS WP), 1,2,2)= * histconOstate,INEW(INSWP),l,2,2)+l

else

hist(jstate,INEW(INSWP), 1,2,2)= * histGstate,rNEW(INSWP),l,2,2)+l

endif

if(tdim) then

c Subtract out contribution of IOLD(INSWP) for all lags in positive z


else izw=iz+l

endif

213 jstate=grid(ix,iy,izw)

if(cond(ix,iy,izw)) then

histcon(IOLD(INSWP) jstate, 1,2,3)= * histcon(IOLD(INSWP) jstate, 1,2,3)-1

else

hist(IOLD(INSWP),jstate, 1,2,3)= * hist(IOLD(INSWP) jstate, 1,2,3)-1

endif

if((iz-l).lt.l)then izw=iz-l+isz

else izw=iz-l

endif


if (cond(ix,iy,izw)) then

histcon(jstate,IOLD(INS WP), 1,2,3)= * histcon0state,IOLD(INSWP),l,2,3)-l

else

hist(jstate,IOLD(INS WP), 1,2,3)= * hist0state,IOLD(INSWP),l,2,3)-l

endif

c Add in contribution of inew for all lags in positive z


else izw=iz+l

endif


214 if(cond(ix,iy,izw)) then

histcon(INEW(INSWP),jstate, 1,2,3)= histcon(INEW(INS WP) jstate, 1,2,3)+1

else

hist(INEW(INS WP) jstate, 1,2,3)= hist(INEW(INSWP),jstate, 1,2,3)+l

endif

if((iz-l).lt.l)then izw=iz-l+isz

else izw=iz-l

endif


if(cond(ix,iy,izw)) then

histcon(jstate,INEW(rNSWP), 1,2,3)= * histcon(jstate,INEW(TNSWP),l,2,3)+l

else

hist(jstate,INE W(INS WP), 1,2,3)= * hist(istate,rNEW(INSWP),l,2,3)+l

endif

endif

800 CONTINUE

888 return end

subroutine thrphist(hist,histcon,grid,isx,isy,isz,cond, * tdirrutau)

C This routine calculates the global three-point histogram of an image. C Calculation is edge-wrapped to minimize boundary effects. The histogram C is stored in an array (ij,k,2,id), where i,j,k=state (1 to im) C at x-itau, x, and x+1. In the horizontal, id=l,2 c and in the vertical up and down when id=3. Q**********************************************************************

integer isx,isy,isz,itau,istate,jstate,kstate integer hist(6,6,6,2,3),histcon(6,6,6,2,3) integer grid(500,500,l),ix,iy,iz,ixw,iyw,izw logical tdim,cond(500,500,l)

c Calculate the Global Histogram

do 100, iz=T,isz do 100, iy=l,isy do 100, ix=l,isx

istate=grid(ix,iy,iz) if(cond(ix,iy,iz)) then

histcon(istate,istate,istate, 1,1)= * histcon(istate,istate,istate, 1,1)+1

else

hist(istate,istate,istate, 1,1)= * hist(istate,istate,istate,l,l)+l

endif

100 continue

c Calculate histograms in ndir=l :positive x

do 200, iz=l,isz do 200, iy=l,isy do 200, ix=l,isx

c Edge wrap if necessary

if((ix+l).gt.isx) then ixw=ix+l+isx

else ixw=ix+l

endif

216 if((ix-itau).lt.l) then

ixww=ix-itau+isx else

ixww=ix-itau endif

istate=grid(ixww,iy,iz) jstate=grid(ix,iy,iz) kstate=grid(ixw,iy,iz)

if(cond(ixww,iy,iz).or.cond(ix,iy,iz).or. * cond(ixw,iy,iz)) then

histcon(istate,jstate,kstate,2,1 )= * histcon(istate,jstate,kstate,2,1)+1

else hist(istate,jstate,kstate,2,1 )=

* hist(istate,jstate,kstate,2,1)+1 endif

200 continue

C Calculate histograms in ndir=2:positive y


C Edge wrap if necessary


else iyw=iy+l

endif

if((iy-itau).lt.l)then iyww=iy-itau+isy

else iyww=iy-itau

endif

istate=grid(ix,iyww,iz) jstate-grid(ix,iy,iz)

kstate=grid(ix,iyw,iz)

if(cond(ix,iyww,iz).or.cond(ix,iy,iz).or. * cond(ix,iyw,iz)) then

histcon(istate,jstate,kstate,2,2)= * histcon(istate,jstate,kstate,2,2)+1

else hist(istate,jstate,kstate,2,2)=

* hist(istate,jstate,kstate,2,2)+l endif

400 continue

if(tdim) then

C Update histograms in ndir=3:positive z

do 600, ix=l,isx do 600, iy=l,isy do 600, iz=l,isz

if((iz-itau).lt.l)then izww=iz-itau+isz

else izww=iz-itau

endif

istate=grid(ix,iy,izww) jstate=grid(ix,iy,iz)


else izw=iz+l

endif

kstate=grid(ix,iy,izw)

if(cond(ix,iy,izww).or.cond(ix,iy,iz).or. * cond(ix,iy,izw)) then

histcon(istate,jstate,kstate,2,3)= * histcon(istate,jstate,kstate,2,3)+l

218 else

hist(istate,jstate,kstate,2,3)= * hist(istate,jstate,kstate,2,3)+l

endif

600 continue

endif

return end

c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

subroutine thrupdate(isx,isy,isz,iold,inew, * tdim,ix,iy,iz,cond,grid,hist,histcon,itau)

c This subroutine updates a three-point histogram in positive and negative c x,y,z directions. The histogram template represents a second-order Markov c chain with P{X}|P{X-l},P{X-itau}. c c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

integer ix,iy,iz,ixw,iyw,izw,ixww,iyww,izww,istate,jstate logical tdim, cond(500,500,l) integer hist(6,6,6,2,3),histcon(6,6,6,2,3) integer grid(500,500,l)

c Consider that a point at grid(ix,iy,iz) has been perturbed by c sample and replacement. Update the three-point histograms

c Adjust the Global proportions

hist(inew( 1 ),inew( 1 ),inew( 1), 1,1 )=hist(inew( 1 ),inew( 1), * inew(l),l,1)4-1

hist(iold( 1 ),iold( 1 ),iold( 1), 1,1 )=hist(iold( 1 ),iold( 1), * iold(l),l,l)-l

c Subtract out the contribution of iold(l) for position x-itau, edge wrap c as required, add in the contribution of inew(l) for the same position, c positive x

Edge wrap if necessary

if((ix+itau).gt.isx) then ixw=ix+itau-isx

else ixw=ix+itau

endif

jstate=grid(ixw,iy,i2;)

if((ix+itau+l).gt.isx) then ixww=ix+itau+1 -isx

else ixww=ix+itau+1

endif

kstate=:grid(ixww,iy,iz)

if(cond(ix4y,iz).or.cond(ixw,iy,iz).or. cond(ixww,iy,iz)) then

histcon(iold( 1 ),jstate,kstate,2,1 )= histcon(iold( 1 ),jstate,kstate,2,1)-1 histcon(inew( 1 ),jstate,kstate,2,1 )= histcon(inew( 1 ),jstate,kstate,2,1)+1

else hist(iold( 1 ),jstate,kstate,2,1 )= hist(iold( 1 ),jstate,kstate,2,1)-1 hist(inew( 1 ),jstate,kstate,2,1 )= hist(inew( 1 ),jstate,kstate,2,1)+1

endif

Subtract out the contribution of iold(l) for position x, edge wrap as required, add in the contribution of inew(l) for the same position, positive x

if((ix+l).gt.isx) then ixw=ix+l-isx

else ixw=ix+l

endif kstate=grid(ixw,iy,iz)

if((ix-itau).lt.l)then

220 ixww=ix-itau+isx

else ixww=ix-itau

endif

istate=grid(ixww,iy,iz)

if(cond(ix,iy,iz).or.cond(ixw,iy,iz).or. * cond(ixww,iy,iz)) then

histcon(istate,iold( 1 ),kstate,2,1) * =histcon(istate,iold( 1 ),kstate,2,1)-1

histcon(istate,inew( 1 ),kstate,2,1) * =histcon(istate,inew( 1 ),kstate,2,1 )+l

else hist(istate,iold( 1 ),kstate,2,1 )=

* hist(istate,iold(l),kstate,2,l)-l hist(istate,inew( 1 ),kstate,2,1 )=

* hist(istate,inew( 1 ),kstate,2,1)+1 endif

c Subtract out the contribution of iold(l) for position x+n, edge wrap c as required. Add in the contribution of inew(l) for the same position, c positive x

if((ix-l-itau).lt.l)then ixw=ix-1 -itau+isx

else ixw=ix-l-itau

endif

istate=grid(ixw,iy,iz)

if((ix-l).lt.l)then ixww=ix-l+isx

else ixww=ix-l

endif

jstate=grid(ixww,iy,iz)

if(cond(ix,iy,iz).or.cond(ixw,iy,iz).or. * cond(ixww,iy,iz)) then

histcon(istatejstate,IOLD( 1 ),2,1 )= * histcon(istate,jstate,IOLD( 1 ),2,1)-1

histcon(istate jstate,INE W( 1 ),2,1 )= * histcon(istate,jstate,INEW( 1 ),2,1)+1

else hist(istatejstate,IOLD( 1),2,1)=

* hist(istate,jstate,I0LD(l),2,l)-l hist(istate,jstate,INE W( 1 ),2,1 )=

* hist(istatejstate,INEW(l),2,l)+l endif

Subtract out the contribution of iold(l) for position y-itau, edge wrap as required. Add in the contribution of inew(l) for the same position, positive y

if((iy+itau).gt.isy) then iyw=iy+itau-isy

else iyw=iy+itau

endif


if((iy+itau+l).gt.isy) then iyww=iy+itau+1 -isy

else iyww=iy+itau+l

endif kstate=grid(ix,iyww,iz)

iAcond(ix,iy,iz).or.cond(ix,iyw,iz).or. * cond(ix,iyww,iz)) then

histcon(IOLD( 1 ),jstate,kstate,2,2)= * histcon(IOLD( 1 ),jstate,kstate,2,2)-1

histcon(INE W( 1 ),jstate,kstate,2,2)= * histcon(INEW(l),jstate,kstate,2,2)+l

else hist(IOLD( 1 ),jstate,kstate,2,2)=

* hist(IOLD( 1 ),jstate,kstate,2,2)-1 hist(INEW( 1 ),jstate,kstate,2,2)=

* hist(INEW(l)jstate,kstate,2,2)+l endif

c Subtract out the contribution of iold(l) for position y, edge wrap c as required. Add in the contribution of inew(l) for the same position, c positive y.


else iyw=iy+l

endif kstate=grid(ix,iyw,iz)

if((iy-itau).lt.l)then iyww=iy-itau+isy

else iyww=iy-itau

endif istate=grid(ix,iyww,iz)

if(cond(bc,iy,iz).or.cond(ix,iyw,iz).or. * cond(ix,iyww,iz)) then

histcon(istate,IOLD( 1 ),kstate,2,2)= * histcon(istate,IOLD( 1 ),kstate,2,2)-1

histcon(istate,INE W( 1 ),kstate,2,2)= * histcon(istate,INEW( 1 ),kstate,2,2)+1

else hist(istate,IOLD( 1 ),kstate,2,2)= hist(istate,IOLD( 1 ),kstate,2,2)-1 hist(istate,INEW( 1 ),kstate,2,2)= hist(istate,INE W( 1 ),kstate,2,2)+1

endif

c Subtract out the contribution of iold(l) for position y+n, edge wrap c as required. Add in the contribution of inew(l) for the same position, c Positive y

if((iy-l-itau).lt.l)then iyw=iy-1 -itau+isy

else iyw=iy-l-itau

endif istate=grid(ix,iyw,iz)

if((iy-l).lt.l)then iyww=iy-l+isy

else iyww=iy-l

endif jstate=grid(ix,iyww,iz)

if(cond(ix,iy,iz).or.cond(ix,iyw,iz).or. cond(ix,iyww,iz)) then

histcon(istate,jstate,IOLD( 1 ),2,2)= histcon(istatejstate,IOLD( 1 ),2,2)-1 histcon(istate,jstate,INEW( 1 ),2,2)= histcon(istate,jstate,INE W( 1 ),2,2)+1

else hist(istatejstate,IOLD( 1 ),2,2)= hist(istate,jstate,IOLD( 1 ),2,2)-1 hist(istate,jstate,INE W( 1 ),2,2)= hist(istate jstate,INE W( 1 ),2,2)+1

endif

Subtract out the contribution of iold(l) for position Z-itau, edge wrap as required. Add in the contribution of inew(l) for the same position. Positive Z

if(tdim) then

if((iz+itau).gt.isz) then izw=iz+itau-isz

else izw=iz+itau

endif


if((iz+itau+1). gt. isz) then izww=iz+itau+1 -isz

else izww=iz+itau+l

endif

kstate=grid(ix,iy,izww)

if(cond(ix,iy,iz).or.cond(rx,iy,izw).or. cond(rx,iy,izww)) then

histcon(IOLD( 1 ),jstate,kstate,2,3)I=

histcon(iold( 1 ),jstate,kstate,2,3)-1

else

224 histcon(inew( 1 ),jstate,kstate,2,3)~ histcon(inew(l),jstate,kstate,2,3)+l

hist(iold( 1 ),jstate,kstate,2,3)= hist(iold( 1 ),jstate,kstate,2,3)-1 hist(inew( 1 ),jstate,kstate,2,3)=

hist(inew(l),jstate,kstate,2,3)+l endif

c Subtract out the contribution of iold(l) for position Z, edge wrap c as required. Add in the contribution of inew(l) for the same position, c Positive 2

if((iz-itau).lt.l)then izw=iz-itau+isz

else izw=iz-itau

endif istate=grid(ix,iy,izw)

if((iz+l).gt.isz) then izww=iz+l-isz

else izww=iz+l

endif kstate=grid(ix,iy,izww)

if(cond(rx,iy,iz).or.cond(bc,iy,izw).or. * cond(ix,iy,izww)) then

histcon(istate,iold( 1 ),kstate,2,3)=

* histcon(istate,iold(l),kstate,2,3)-l histcon(istate,inew(l),kstate,2,3)=

* histcon(istate,inew( 1 ),kstate,2,3)+1 else

hist(istate,iold(l ),kstate,2,3)=

hist(istate,iold( 1 ),kstate,2,3)-1 hist(istate,inew( 1 ),kstate,2,3)= hist(istate,inew( 1 ),kstate,2,3)+l

endif Endif return end

program prephist

c This program reads in a first or a second order Markov chain, c The input file has a header line and then the m*m matrix in c m rows with m columns each.

c If it's a first order chain, the chain is powered to get the c stable probability vector. If its a second order Markov chain, c a ID simulation is made in order to get the stable probability c vector approximated.

c Then the user inputs in the size of an equidimensional 2D field c to process.

c The program then makes the appropriate input file for manneaLf

c Non-directionality is not assumed. For a 1st order matrix, c the backwards tally matrix approach is used to determine the c reversed Markov histogram. For the 2nd order matrix, the reversed c statistics are read from the ID simulation.

integer iord,isz,isx,isy,idim logical tdim

c The user has to enter the order of the matrix. The order can c be 1 or 2. This is synonomous with single and double dependency.

print *,'Enter the dependency of the matrix (1 or 2)' print * read *, iord print *, 'Enter the x-size of the square field for processing.'

201 print *, 'Maximum is 500.' print * read *, isx

if(isx.gt.500) then print *, 'Enter again. You entered ',isx go to 201

endif

print *, 'Enter the y-size of the square field for processing.' 202 print *, 'Maximum is 500'

print * read *, isy

if(isy.gt.500) then print *, 'Enter again. You entered ',isx go to 201

endif

c The user is prompted to enter the dimensionality of the grid c to be processed by manneal. Either two or three.

print *, 'Enter 2 if two-dimensional.' print *, 'Enter 3 if three-dimensional' print *

13 read *, idim

tdim=.false.

if(idim.eq.3) then tdim=.true.

endif

if(idim.lt.2.0R.idim.gt.3) then print * print *, 'Please enter either 2 or 3.' goto 13

endif

if(idim.eq.3) then print * print *, 'Enter the z-size of the square field for processing .' print * read *, isz

else isz=l

endif

if(iord.eq.l) then call firstord(iord,isx,isy,isz,tdim)

endif

if(iord.eq.2) then call secondord (isx,isy,isz,iord,tdim)

endif end

227

subroutine firstord(iord,isx,isy,isz,tdim)

c * This subroutine reads in a first single dependency Markov matrix * c * from a file called matrix.in. A parameter file for Manneal that * c * contains the derived two point histograms for 2 (2D) or 3 (3D) * c * orthogonal axes. Directionality is accomodated implicitly in the * c * calculation of the reverse two point histograms because the * c * calculation essentially produces a tally matrix with im*im * c * transitions. The tally matrix can be transposed to get the * c * reverse directon. * c * * c * In this version, the writing of the reverse matrices is * c * disengaged. The version of manneal only enforces the forward * c * transition matrices since they automatically create the backward. *

real matrix(6,6),power(6,6,100) integer im,isz,iord,isx,isy,ansav logical tdim

C initialize the arrays

do 10, i=l,6 do 10,j=l,6 do 10, k=l,100

power(i,j,k)=0 matrix(ij)=0

10 continue

c Read in the matrix from the file matrix.in. Format is free. c The first line of the file contains imFnumber of states. c Thereafter in the file there should be m lines with c m columns. These are the elements of the Markov transition matrix.

open(3 ,file-matrix, in') read(3,*)im

if(im.gt.6) then print * write(*,*) 'Warning: Number of States Exceeds 6.' write(*,*) 'Check your input file. Program terminated.'

go to 999

endif

print * print *, 'The Markov transition matrix from file is:' print *

do 20, i=l,im read(3,*) (matrix(i,j),j=l,im) write(*,*) (matrix(i,j),j=l,im)

20 continue close(3)

c To calculate the stable probability vector, power the matrix c 100 times. Usually this is sufficient to reach the stable c independent probabilities. If not, this part of the code needs c to be altered.

do 25, i=l,im do 25, j=l,im

power(i,j,l)= matrix(i,j) 25 continue

C Power the matrix up to 100 to get stable proportions.

do 30, ip=2,100 do 30, i=l,im do 30, j=l,im do 30, k=l,im

power(i,j,ip)= power(i,j,ip)+power(i,k,ip-1 )*matrix(k,j) 30 continue

print *

c Do a quick error check. The stable probability vector should c add to 1 within roundoff error.

sumstate=0.0 do 100,i=l,im

write(*,*) 'The proportion of State: ',i,' is',power(l,i,ip-l) sumstate=sumstate+po wer( 1 ,i,ip-1)

100 continue print * write(*,*) 'The sum of the proportions is: ',sumstate

if(sumstate.lt.0.99.or.sumstate.gt. 1.01) then

write(*,*) 'This sum should equal 1.000.' write(*,*) 'This sum is outside acceptable range.' write(*,*) 'Go back and check your input matrix or consider',

* 'altering the code at line 139.' write(*,*) 'Program terminating.' go to 999

endif

c Prepare the input file for manneal.

8 open(8,file-'markov.parm') 101 format(il0,10x,a40) 103 format(6fl0.3) 102 format(fl0.3,10x,a20)

write(8,101) im dumber of states and Proportions.' write(8,*) (power(l,i,ip-l),i=l,im) write(8,101) iord,'Dependency or order of matrix' write(8,101) isx,'X dimension of grid' write(8,101) isy,'Y dimenson of grid'

if(tdim) then write(8,101)1 /Dimension flag, 1 = 3D' write(8,101) isz,'Z dimension of grid'

else write(8,101) 0,'Dimension flag, 0=2D'

endif

write(8,101) 0, 'Condition flag, edit if conditioning.'

If(.not.tdim) then

c Write to the Manneal parameter file the two point histograms c for 2 directions. Direction 1 is +x, direction 2 is +y.

c The number of transitions from one state to anouther is determined c by multiplying P(i)*P(j|i)*total number of transitions = x dimension c times y dimension because Manneal uses edge wrapping to reduce edge c effects. The code reads: c 1. (power( 1 ,i,ip-1) = the independent probability of c state i (ip-1 being the power 100, ip last being=T01) c 2. power(i,j,l) = the transition probability from any state c i to j if direction is 1 or 3. If direction is 2 or 4, use c the transition probability from j to i. This can be done c only for single dependent matrices.

230 c 3. isx = dimension of 2D grid in the x dimension, c 4. isy = dimension of 2D grid in they dimension, c The rest of the parameters are printed to help in debugging output.

write(*,*) 'Do you want to remove directionality in the ', * 'horizontal i.e., x direction, by averaging off diagonals?'

write(*,*) 'Enter 0 for no, 1 for yes.' print * read *, ansav

if(ansav.eq.O) then print *,'Ansav= 0, No averaging' do 50, i=l,im do 50, j=l,im

write(8,*) nint(power( 1 ,i,ip-1 )*power(ij, 1 )*isx*isy),i,j, 1 c write(8,*) nint(power(l j,ip-l)*power(j,i,l)*isx*isy),i,j,2

write(8,*) nint(power( 1 ,i,ip-1 )*power(i,j, 1 )*isx*isy),i,j,2 c write(8,*) nint(power(l,j,ip-l)*power(j,i,l)*isx*isy),i,j,4 50 continue

else print *,'Ansav = 1, Average Offdiagonals' do 51, i=l,im do 51, j=l,im

write(8,*) nint( (power(l,i,ip-l)*power(i,j,l)/2 + * power( 1 ,j,ip-1 )*power(j,i, 1 )/2)*isx*isy),i,j, 1

c write(8,*) nint( (power(l,i,ip-l)*power(i,j,l)/2 + c * power(l,j,ip-l)*power(j,i,l)/2)*isx*isy),i,j,2

write(8,*) nint(power( 1 ,i,ip-1 )*power(i,j, 1 )*isx*isy),i,j,2 c write(8,*) nint(power(l,j,ip-l)*power(j,i,l)*isx*isy),i,j,4 51 continue

endif Endif

If(tdim) then

c For three dimensional cases, edge wrap in the vertical only if c there are more than five layers. Otherwise the field will be c correlated unto itself close to the midpoint of the field.

do 60, i=l,im do 60, j=l,im

write(8,*) nint(power( 1 ,i,ip-1 )*power(i,j, 1 )*isx*isy*isz), * ij,l

c write(8,*) nint(power(l,j,ip-l)*power(j,i,l)*isx*isy*isz),

c * i,j,2 write(8,*)nint(power(l,i,ip-l)*power(i,j,l)*isx*isy*isz), U,2

c write(8,*) nint(power( 1 ,j,ip-1 )*power(j,i, 1 )*isx*isy*isz), c * i,j,4

write(8,*) nint(power( 1 ,i,ip-1 )*power(i,j, 1 )*isx*isy*isz), * U,3

c write(8,*) nint(power(lj,ip-l)*power(j,U)*isx*isy*isz), c * i,j,6 60 continue

if(isz.lt.5) then print * write(*,*) 'Warning:' write(*,*) 'Your vertical thickness is less than 5 units.',

* 'Edge wrapping is used in mannealing. You may be correlating', * 'over .25 of the distance across the field in the vertical.', * 'Edge effects could be severe.'

print * write(*,*) 'Program will continue but you should consider',

* 'adding more layers and rerunning the program.' endif Endif

close(8)

write(*,*) 'An unconditioned markov.parm file has been created.'

999 return

End

subroutine secondord (isx,isy,isz,iord,tdim)

c A program to create a ID sequence with a second-order c probability structure and write a multipoint c histogram to a markov.parm file. When itau=l, a transpose c can be used to calculate the backward histograms. When c itau>l, the simulation is used to calculated the backward c histograms.

realprob(6,6,6),cumprob(6,6,6),numstate(6), * propstate(6), * probf(6,6,6),probb(6,6,6), * margpf(6,6),margpb(6,6),summatf(6),summatb(6), * sumrowf(6,6),sumrowb(6,6),tallyf(6,6,6), * tallyb(6,6,6)

integer histf(6,6,6),histb(6,6,6), * im, itime,itau,obsvat(10013),isz,iord, * isx,isy

logical tdim

external ranO

c Initialize the arrays

do 5, i=l,6 do 5, j=l,6 do5,k=l,6 do 5, ii=l,6

prob(i,j,k)=0 cumprob(i,j ,k)=0 numstate(i)=0 propstate(i)=0 margpf(ij)=0 margpb(i,j)=0 summatf(i)=0 summatb(i)=0 probf(i,j,k)=0 probb(ij,k)=0 sumrowf(i,j)=0 sumrowb(i,j)=0 tallyf(i,j,k)=0 taUyKi,j,k)=0

5 continue

c initialize the obsvat array to zeros do 6, i=l,10013

obsvat(i)=0 6 continue

c Seed the simulation with two states (the minimum).

obsvat(l)=l obsvat(2)=2 obsvat(3)=l obsvat(4)=2 obsvat(5)=l obsvat(6)=2

c Set a random number seed

iy=56799

c Set a counter to use with the ID simulation, c This should equal the simulation length minus 1 plus 6. c A simulation length of 10000 is used here. If this is c altered, remember to redimension the array obsvat.

itime=10005

c The ID simulation held in the array obsvat will have c 10,000 simulated values in cells 7 to 10,006. c Read in simulation parameters and matrices c These are in the file matrix.in. The first line contains c the parameter im = the number of states. c The next parameter is itau, the lag that characterizes c the spatial scale of a Markov double dependency. If the c state of a Markov chain at X is dependent upon a prior state A c and a state B prior to that, then itau is the lag separating c A and B while imu is the lag separting A and X. Imu is implicitly c assumed to be one here. If imu>l, there would be some sort of effect c like a nugget in conventional geo statistics. c After the header line, there should be im*im rows with im columns. c These are the double dependent transition matrices. The first c im rows represent P(k[j|i=l), the second im rows represent P(k,j|i=2) c and so on up to P(k[j|i=im). No lines separate the im groups of im rows c with im columns. Format is free format.

open(3,file='matrbc.in') read(3,*) imjtau if(im.gt.6) then

write(*,*) "Number of states exceeds 6.' write(*,*) 'Program stopped.' go to 999

endif

234 if(itau.ge.6) then

write(*,*) 'Itau is greater than 5.' write(*,*) 'Program stopped.' go to 999

endif

c Read in the probabilities. Note that P(k|j|i) is stored as c prob(i,j,k).

do 10, i=l,im do 20,j=l,im

read(3,*) (prob(i,j,k),k=l,im) 20 continue 10 continue

close(3)

Prepare cumulative probabilities for use in simulation.

do 25 i=l,im do 25 j=l,im

cumprob(i,j, 1 )=prob(i,j, 1) continue

do 30, i=l,im do 30,j=l,im do 30, k=2,im

cumprob(i,j,k)=cumprob(i,j,k-1 )+prob(i,j,k) 30 continue

C Perform the simulation

do 40, istep=6,itime i=obsvat(istep-itau) j=obsvat(istep) x=ranO(iy) do 50, k=im,l,-l

if(x.le.cumprob(i,j,k)) then obsvat(istep+1 )=k

endif continue

continue

C

25

50 40

C Determine the global histogram of the simulation

instepH)

do 65, i=7,itime+l instep=instep+l

do 65, istate=l,im if(obsvat(i).eq.istate) then

numstate(istate)=numstate(istate)+1 endif

65 continue print * write(*,*) 'The number of simulated values is:',instep print * do 68, i=l,im

propstate(i)=(numstate(i))/(instep) write(*,*) 'Proportion of state ',i,' is:',

* propstate(i),numstate(i) 68 continue

c Determine from the simulation the forward and backwardtransition matrices c matrices from n=l to in. The tallys are recorded in an arrays tallyf(i,j,k) c and tallyb(i,j,k). The transition matrices are calculated so the c three-point histograms for different sized grids can be generated.

c Calculate the forward tally matrices

do 200, istep=7,itime-itau istate=obsvat(istep) jstate=obsvat(istep+itau) kstate=:obsvat(istep+itau+1) talfyf(istate,jstate,kstate)=

* tallyf(istate,j state,kstate)+1 200 continue

c calculate the backward tally matrices

do 201, istep=itime+l,8+itau,-l istate=obsvat(istep) jstate=obsvat(istep-itau) kstate=obsvat(istep-itau-1) tallyb(istate,jstate,kstate)=

* tallyb(istate,jstate,kstate)+l

236 201 continue

c calculate the rowsums

do 220, ii=l,im do 220, ij=l,im do 220, ik=l,im

sumrowf(ii,i))=sumrowf(ii,ij)+ * tallyf(ii,ij,ik)

sumrowb(ii,ij)=sumrowb(ii,ij)+ * tallyb(ii,ij,ik)

220 continue

c calculate the transition probabilities


if(sumrowf(ii,ij).gt.O) then probf(n,ij,ik)=real(tallyf(ii,ij,ik)/sumrowf(ii,ij))

else probf(ii,ij,ik)=0

endif if(sumrowb(ii,ij).gt.O) then

probb(ii,ij,ik)=real(tallyb(ii,ij,ik)/sumrowb(ii,ij)) else

probb(ii,ij,ik)=0 endif

225 continue

c calculate the marginal probabilities = P(j(x)|i(x-itau))

do 230, ii=l,im do 230, ij=l,im

summatf(ii)=summatf(ii)+sumrowf(ii,ij) summatb(ii)=summatb(ii)+sumrowb(ii,ij)

230 continue

do 205, ii=l,im do 205, ij=l,im

margpf(ii,ij)=sumrowf(ii,ij)/summatf(ii) margpb(ii,ij)=sumrowb(ii,ij)/summatb(ii)

205 continue

237 open(33,file-debug.out') write(33,*) 'Forward Tally Matrix' do 710, i=l,im

write(33,*) i do 710,j=l,im

write(33,*) (tanyf(i,j,k),k=l,im),sumrowf(i,j) 710 continue

write(33,*) write(33,*) 'Forward Transition Matrix' do 712, i=l,im


write(33,*) (probf(i,j,k),k=l,im) 712 continue

write(33,*) write(33,*) 'Forward Marginal Probabilities' do 714, i=l,im

write(33,*) 'Sum of all row sums for state:',i'is', * summatf(i)

do 714, j=l,im write(33,*) i,j, margpf(i,j)

714 continue

write(33,*) write(33,*) 'Backward Tally Matrix' do 711, i=l,im


write(33,*) (tallyb(i,j,k),k=l ,im),sumrowb(i,j) 711 continue

write(33,*) write(33,*) 'Backward Transition Matrix' do 713, i=l,im

write(33,*) i do 713, j=l,im

write(33,*) (probb(Lj,k),k=l,im) 713 continue

write(33,*) write(33,*) 'Backward Marginal Probabilities' do 715, i=l,im write(33,*) 'Sum of all row sums for state:',i,'is',

* summatb(i) do 715,j=l,im

write(33,*) i,j, margpb(i,j) 715 continue

close(33)

c calculate the forward and backward histograms

if (tdim.AND.isz.lt.(5-itau)) then print * write(*,*) 'You are using too few layers to ',

* 'allow edge wrapping.' write(*,*) 'Please reconsider the vertical',

* 'dimension of your model' endif


histf(h\ij,ik)==nint(propstate(ii)*rnargpf(ii,ij) * *probf(ii,ij,ik)*isx*isy*isz)

histb(ii,ij,ik)=nint(propstate(ii)*rnargpb(ii,ij) * *probb(ii,ij,ik)*isx*isy*isz)

300 continue

open(8,file='markov.pariri) 101 format(il0,10x,a40) 102 format(fl0.3,10x,a40) 103 format(6fl0.3)

write(8,101) im dumber of states and Proportions' write(8,*) (propstate(i),i=l,im) write(8,101) iord,'Dependency or Order of matrix' write(8,101) itau,'Value of itau' write(8,101) isx,'X dimension of grid' write(8,101) isy,'Y dimension of grid'

if(tdim) then write(8,101) 1 ,'Dimension flag, 1=3D' write(8,101) isz ,'Z dimension of grid'

else write(8,101) 0,'Dimension flag, 0=2D'

endif

write(8,101) 0,'Conditioning Flag. Edit if conditioning.' do 320, ii=l,im do 320, ij=l,im do 320, ik=l,im

write(8,*) histf(ii,ij,ik), ii,ij,ik,l c write(8,*) histb(ii,ij,ik), ii,ij,ik,2

write(8,*) histf(ii,ij,ik), ii,ij,ik,2 c write(8,*) histb(ii,ij,ik), ii,ij,ik,4

if(tdim) then write(8,*) histf(ii,ij,ik), ii,ij,ik,3

c write(8,*) histb(ii,ij,ik), ii,ij,ik,6 endif

320 continue close(8)

write(*,*) 'An unconditioned markov.parm file has been created.'

999 return end

FUNCTION ranO(idum)

c This subroutine has been removed for copywrite.

Sample of formatted parameter file MARKOV.PAR created by prephist.for.

0.721045E 1

100 100 0 0 646 646 33 33 25 25 8 8 8 8 13 13

2164 2164

70 70

294 294 51 51 31 31 62 62

856 856 109 109 16 16 6 6

270

Number of states and Proportions. •01 0.259156 0.107416 0.463543

Dependency or order of matrix X dimension of grid Ydimensonofgrid

Dimension flag, 0=2D Condition flag, edit if conditioning.

0.977788E-01

2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4

1 1 2 2 3 3 4 4 5 5 1 1 2 2 3 3 4 4 5 5 1 1 2 2 3 3 4 4 5 5 1 1 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1

270 4 2 2 123 4 3 1 123 4 3 2 4175 4 4 1 4175 4 4 2 61 4 5 1 61 4 5 2 25 5 1 1 25 5 1 2 62 5 2 1 62 5 2 2 0 5 3 1 0 5 3 2 49 5 4 1 49 5 4 2 842 5 5 1 842 5 5 2

Sample of formatted parameter file ANNEAL.PAR used by mannealfor

annealing schedule 1. .1 50000 500000 0 0.00100 1000 1 1.0 1.0 0.0 2 1 2 734379

Initial temperature Cooling parameter or lambda Required number of acceptances at T Maximum number of perturbatations at T Number of excess trials to reduce temperature Convergence criterion Report to screen after this many perturbations Number of realizations to generate. wl, the weight on the offdiagonals w2, the weight on the diagonals w3, the weight on the penalty term swapflag: 1 replacement, 2=swap metflag: l=true annealing, 2=iterative improvement initflag:l=random prop.,2=strict prop.,3=out.fil a large odd integer for the random number seed

243

Appendix C

Representative Core Photographs, Gloucester Borehole Logs, and

Laboratory-Measured Hydraulic Data

244

%

5\

*

z BE

Plate C.l: Representative photograph of coarse sand lithotype.

245

a

tflm flittf I ' 0 O

o J |'^gf^§J c

3^ i Sifil^J mm* s i ^ 3 ^ •'•^jr^^''*'" -^ a o 3

CV I s^^." ,^^.^£ sT^.^ -„ ..^,* "&

^ ^ ^-•*,--v"--"^/-v<J V -fS ^ ^ ^-•*,--v"--"^/-v<J V - T>

-J V j > - * * - . ' ^ i * . ' . * ' * . * . ^ <''."^ ' • '- ; 0 ** • ' " ' • - A " „ " ' •'•'K- - [ 8 cc K

J& c-VC:~ •-'-' " '''** " ' N ' "-

** m <. • •%." T^ie^yjt; 0 M

•c

K ^ * * ' / /r^_ -« **T1["*; •

00 I 0 [ * tl W\i

Ui - '•.••"/ '••. ..'•" " f " ? • ' ~ • . » • " r, * , ! D

m « 1 "'} Wj

•"^ ff-;?, -K i " 'V^ i r '"*~i>" 0 <

r v g? ;v" '^JVx'" OJ 2

<f e^"»5»-",' f ->.- '.- '•--."^t- J-" » ^j 0£ 'rr;--^ ^ < j a <

<A tills ?!§£'• > J E *"— £

^p l £ . ." '""^"ki. jt,t, A . > : j f l -—a > c

• **IBBr «, 3

Plate C.2: Representative photograph of fine sand lithotype. Note clay interbed

just above midpoint.

246

5*

c

as

c o 3

£ Hi

4™

- * V ^ '

Plate C.3: Representative photograph of silt lithotype. Here bedding is very disturbed

and intermixed with silty clay lithotype.

247

- ' • ' •

^0

2 U

O

f1

- - £

Plate C.4: Representative photograph of silty clay lithotype. Note fine sand stringers

near top and base.

248

HH

2 o

OS

s ° •3

f

Plate C.5: Representative photograph of stiff clay lithotype. Note fine sand stringers

in lower half of core.

249

3

Plate C.6: Representative photograph of diamict lithotype. Here diamict (upper half ofcore)

is overtop a silty clay with a sharp basalcontact.

250

Medium to coarse sand

fine sand

Silt

Silty clay

Clay

Diamict

Unrecovered section

Lithology legend for borehole logs.

Drilled Depth

(m)

8

Drillhole UC95-2 Lithology Logarithm of Kv (m/s)

Northing: 20.54 m Easting: 68.90 m Elevation: 92.27 m

Porosity

251

o o co r-v. ,o m *q- D O D O D O O i - N (0 ^j iO d> d> c> cS d d

• »

Drilled Depth

(m)

11

Drillhole UC95-2 Northing: 20.54 m Easting: 68.90 m Elevation: 92.27 m

252

Llthology Logarithm of Kv (m/s) Porosity O Ch to rv •O uj> ^ Q O O O O

o — N 03 9 i o" d a 6 d

12

13

14

Drilled Depth

(m)

Drillhole UC95-2

Lithology Logarithm of Kv (m/s)


Porosity

253

CM ~~ o o CO r^ •9 "9 "7 o o o o o Q o CN CO

d

15

16

0 0 o

° ,_0_ o

\ /

\ /

\

1 1

Drilled Depth

(m)

Drillhole UC95-3,

Lithology

A \J

7

\

/ \ / \

/

o

Drilled Depth

(m)

8

Northing: 50.17 m Easting: -23.97 Elevation: 99.94 m Lithology

254

10

Drillhole UC95-3

Drilled Depth

(m)

11 Lithology

Drilled Depth

(m)

Northing: 50.17 m Easting:-23.97 m Elevation: 99.94 m

Lithology

255

12

13

14

Drillhole UC95-4

Drilled Depth

(m) Lithology

Drilled Depth

(m)

11

Northing:-128,40 m Easting: 18.51 m Elevation: 98.01 m

Lithology

256

Drillhole UC95-4

Drilled Depth

(m)

Northing:-128.40 m Easting: 18.51 m Elevation: 98.01 m

257

Lithology

Drillhole UC95-5

Drilled Depth

(m) Lithology

7

8

Northing:-80.15 m Easting: 34.85 m

Depth Elevation: 97.43 m (m) — Lithology

o

258

Drillhole UC95-6

Drilled Depth

(m) Lithology

Drilled Depth

(m)

Northing:-26.81 m Easting: 54.01 m Elevation: 97.31 m

Lithology

259

7

8

Drilled Depth

(m)

Drillhole UC95-7


260

Lithology

Logarithm of Kv (m/s)

C N ^ o o a > f ^ o i r > ^ r

Porosity

o o

CM ^J tO

7

Drillhole UC95-7

Drill Depth

(m)

8-



CM r~ o O QO Porosity

o o o o o o o — oj to T io o o o d o o

261

9

10

Drilled Depth

(m)

11 Lithology

Drillhole UC95-7



262

Porosity

o o o o

12

13

14

Drilled Depth

(m)

16


263

Lithology Logarithm of Kv (m/s) Porosity

8 S 8 8 § 8 d d d d d o

Drillhole UC95-7

Drilled Depth

(m)


264

Lithology Logarithm of Kv (m/s) •" o o oo r o ip 7

Porosity

o o o o o o

20

Drillhole UC95-8 Drilled Depth

(m) Lithology

Logarithm of Kv (m/s) O o <p r*. >o

Northing:21.37 m Easting: 13.35 m Elevation: 98.14 m

Porosity <7 o o o o o o • o — CM m -=r io

265

8

Drilled Depth

(m)

Drillhole UC95-8



266

Porosity

Lithology ( N " - o o - o p r ^ - - 0 1 0 * 7

o o o & o o

Drillhole UC95-8 Drilled Depth

(m)


267

11 Lithology

Logarithm of Kv (m/s) o o co r . -o «p -ST

Porosity

12

14

Drillhole UC95-8

Drilled Depth

(m)


268


O O op r s <> up

Porosity

o o o *T U3

o d

16

No measurements taken

17

Drilled Depth

(m)

Drillhole UC95-9

Lithology



C M 1 - o o o p r v - p u p ^ r

Porosity o o o o o O i— N CO "T 0 0 0 0 0

269

6 I —

7

8

Drilled Depth

(m)


270


O O QO r^ -O 10

Porosity

O O O O O D

Drillhole UC95-9

Drilled Depth

(m) Lithology


271

Logarithm of Kv (m/s) Porosity

o o o o o o o d d

> •

Drilled Depth

Cm)

14 Lithology

Drillhole UC95-9


CM ^ o O- co rv. >o


272

Porosity

15

16

17

Drillhole UC95-10 Northing: 29.94 m Easting: 25.30 m

Drilled Depth

(m)

Drilled Depth

(m)

Elevation: 97.97 m

Lithology

273

8 Lithology

7 10

11

Drillhole UC95-10

Drilled Depth

(m)

13

Lithology

Drilled Depth

(m)


274

Lithology

16

Drilled Depth

(m) Lithology

Drillhole UC95-11


D o- of rj. -o

Northing: 21.78 Easting: 42.55 m Elevation: 97.58 m

275

Porosity o o o o o o o i— <N co *q- m o o o o ci o

7

Drilled Depth

(m)


276


o o oo r~ ~ct ir> *rr

Porosity o o o o o o D <- CM CO <7 lO

o o o o o o

Drilled Depth

(m)


277

Lithotypes Logarithm of Kv (m/s)

o o- op r . o w

Porosity o o o o o o O i— CM CO ^T "5

d d d d d o

Drilled Depth

(m) Lithology

Drillhole UC95-11 Logarithm of Kv (m/s)

c \ i ^ o o - o o r > - o i o


278

Porosity

co -^ in

Drillhole UC95-12

Drilled Depth

(m)

8

Lithology


279


o o cp r-> *o m

Porosity

o o o o o o d d d d

• »

Drilled Depth

(m)

9

10

Drillhole UC95-12

Lithology


280

Logarithm of Kv (m/s) o <> cp rv. >o up

Porosity

o o o o o o o •— CM

o d d

CO *ZT i&

o d d

Drilled Depth (m)

Lithology

Drillhole UC95-12

Logarithm of Kv (m/s) O s i ^ o C K c p r j - -o n>

14

Northing: 6.07 m Easting: 665.42 m Elevation: 97,22 m

281

Porosity o o o o o o O r - OJ CO "=7 l O

b 6 o 6 o 6

Drilled Depth (m)

Drillhole UC95-13 Northing: -2.51 m Easting: 62.55 m Elevation: 97.20 m

282


o o- op rj* o Porosity

8 £ 8 8 d d d d

8

7

8

Drilled Depth

(m)

Drillhole UC95-13 Northing: -2.51 m Easting: 62.55 m Elevation: 97.20 m

283


o o- op r -o up Porosity

o o o o o o o -—

Drilled Depth

(m)

13

Drillhole UC95-13

Lithology

Northing: -2.51 m Easting: 62.55 m Elevation: 97.20 m

284

Logarithm of Kv (m/s) o o- cp r . -o ur>

Porosity o o o o o o O - (\J CO q Ifl

o o o o o d

Drillhole UC95-13

Drilled Depth

(m)


285


o o- cp rj. -p up Y

Porosity

b

Drillhole UC95-14

Drilled Depth

On) Lithology

Drilled Depth

(m)


286

Lithology

8

Drillhole UC95-14

Drilled Depth

(m)


287

Lithology

Drilled Depth (m)

7

8

Drillhole UC95-15

Lithology

Drilled Depth (m)

8-


288

Lithology

10

11

Drilled Depth

(m)

11 Lithology

Drillhole UC95-15

Drilled Depth

(m)

14


Lithology

289

12

13

14

Table CI: Summary of Core-Measured K and Porosity Data 290

Elevation of Lithologic Proportions Sample Med. Fine Silty Dia- Vertical

Hole Midpoint (m) Sand Sand Silt Clay Clay mict K(m/s) Porosity 2 86.69 0 0 0 0 0 2.42E-06 0.47 2 86.69 0 0 0 0 0 3.04E-06 2 86.78 0 0 0 0 0 7.21E-06 0.35 2 86.78 0 0 0 0 0 7.42E-06 2 86.89 0 0 0 0 0 1.25E-05 2 86.89 0 0 0 0 0 1.23E-05 0.44 2 86.99 0 0 0 0 0 1.62E-05 2 86.99 0 0 0 0 0 1.58E-05 2 87.09 0 0 0 0 0 2.23E-05 0.49 2 87.09 0 0 0 0 0 2.25E-05 2 87.19 0 0 0 0 0 1.03E-05 2 87.19 0 0 0 0 0 1.15E-05 0.34 2 87.61 0.54 0 0.5 0 0 0 1.51E-07 0.32 2 87.71 0 0.8 0.3 0 0 0 1.79E-06 2 87.71 0 0.8 0.3 0 0 0 1.89E-06 2 87.81 0 0.1 0.9 0 0 0 2.97E-06 0.38 2 87.81 0 0.1 0.9 0 0 0 2.96E-06 2 88.20 2.29E-08 2 88.32 1.56E-08 2 88.46 0 0 0 0 0 1.12E-08 0.43 2 88.61 0 0.0 0 0 0 2 88.86 0 0 0 0 0 0.37 2 88.96 0 0 0 0 0 2.33E-08 0.41 2 89.06 0 0 0 0 0 2.87E-07 2 89.06 0 0 0 0 0 2.65E-07 2 89.17 0 1 0.0 0.0 0 0 3.60E-06 2 89.17 0 1 0.0 0.0 0 0 3.51E-06 2 89.17 0 1 0.0 0.0 0 0 3.46E-06 2 89.34 0 0.2 0 0.8 0 0 6.22E-08 0.43 2 89.46 0 0.3 0 0.7 0 0 2.74E-08 0.45 2 89.56 0 0.4 0.0 0.6 0 0 6.00E-08 2 89.67 0 1 0 0 0 0 4.77E-06 2 89.67 0 1 0 0 0 0 4.67E-06 0.41 2 89.78 0 0.9 0 0 0.1 0 4.15E-07 2 89.78 0 0.9 0 0 0.1 0 4.04E-07 0.41 7 91.80 0 1 0 0 0 0 1.37E-06 7 91.80 0 1 0 0 0 0 1.34E-06 7 91.91 0.74 0.3 0 0 0 0 2.06E-06 7 91.91 0.74 0.3 0 0 0 0 2.05E-06 0.34 7 92.01 1 0 0 0 0 0 1.52E-05 7 92.01 1 0 0 0 0 0 1.57E-05 0.42

7 92.11 1 7 92.11 1 7 92.21 0 7 92.21 0 7 92.31 0 7 92.31 0 7 92.42 0 7 92.42 0 7 92.42 0 7 92.42 0 7 92.52 0 7 92.52 0 7 92.62 7 92.62 7 92.62 7 92.72 7 92.72 7 92.83 0 7 92.92 0.04 7 93.03 0 7 93.03 0 7 93.13 0.09 7 93.13 0.09 7 93.23 0.54 7 93.35 1 7 93.35 1 7 93.53 0 7 93.64 0 7 93.74 0 7 93.84 0 7 93.94 0 7 94.07 0 7 94.17 0 7 94.27 0 7 94.38 0 7 94.38 0 7 94.47 0 7 94.56 0 7 94.85 0 7 95.06 0.83 7 95.16 0 7 95.56 0 7 95.77 0.55 7 95.77 0.55 7 96.08 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.1 0 0.9 0 0.7 0 0 0.0 0 0.4 0.6 0 0 0.4 0.6 0

0.3 0.3 0 0 0.3 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.3 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0.0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0.4 0.6 0 0 0.2 0 0 0 0.2 0.8 0 0 0.4 0.6 0 0 0.5 0 0 0 0.5 0 0 0

1 0 0 0

0 1.92E-05 0 2.04E-05 0.37 0 8.64E-07 0 9.14E-07 0.43 0 1.30E-08 0 1.58E-08 0.45 0 3.02E-06 0.63 0 3.30E-06 0 3.22E-06 0 3.16E-06 0 4.23E-07 0 3.72E-07 0.37 0 1.41E-05 0 1.45E-05 0 1.19E-05 0.50 0 3.31E-06 0 3.43E-06 0 6.54E-08 0.36

0.2 3.51E-07 0.50 0 9.76E-08 0 8.09E-08 0.34

0.3 1.56E-06 0.3 1.59E-06 0.5 1.49E-06 0 7.24E-06 0 6.46E-06 0.38

0.7 4.24E-08 0.38 0 1.86E-08 0 7.64E-09 0.44 0 1.85E-07 0 6.85E-07 0.40 0 6.08E-09 0.35 0 2.68E-06 0.42 0 1.62E-06 0.37 0 1.59E-07 0 2.36E-07 0.31 1 8.91E-08 0.28 1 5.35E-08 0.28 0 4.24E-07 0.34 0 2.31E-06 0.34 0 4.72E-06 0.33 0 4.41E-07 0.49 0 1.87E-07 0 1.96E-07 0.37 0 7.13E-06 0.53

292 7 96.17 0 1 0 0 0 0 1.64E-06 0.42 7 96.28 0 0 1 0 0 0 9.62E-06 7 96.28 0 0 1 0 0 0 9.29E-06 7 96.38 0 0 1 0 0 0 7.23E-06 7 96.38 0 0 1 0 0 0 6.95E-06 0.44 8 93.11 0 0 0 0 0 9.60E-06 8 93.11 0 0 0 0 0 7.48E-06 0.36 8 93.21 0 0 0 0 0 3.40E-06 8 93.21 0 0 0 0 0 8.61E-06 0.37 8 93.31 0 0 0 0 0 2.98E-06 8 93.31 0 0 0 0 0 2.92E-06 0.36 8 93.42 0 0 0 0 0 4.20E-06 8 93.42 0 0 0 0 0 4.02E-06 0.36 8 93.52 0 0 0 0 0 4.30E-06 0.38 8 93.52 0 0 0 0 0 4.31E-06 8 93.80 0 0 0 0 0 9.62E-07 8 93.80 0 0 0 0 0 8.39E-07 0.63 8 93.90 0 0 0 0 0 1.78E-06 8 93.90 0 0 0 0 0 1.04E-06 0.36 8 94.01 0 0 0 0 0 3.04E-07 8 94.01 0 0 0 0 0 3.12E-07 0.34 8 94.12 0 0 0 0 0 3.11E-06 0.33 8 94.12 0 0 0 0 0 3.23E-06 8 94.33 0.06 0.0 0 0 0 0.9 5.16E-09 8 94.33 0.06 0.0 0 0 0 0.9 5.28E-09 0.27 8 94.43 0.16 0.8 0 0 0 0 2.47E-07 8 94.43 0.16 0.8 0 0 0 0 2.48E-07 0.24 8 94.53 0 0 0 0 0 5.32E-06 0.37 8 94.53 0 0 0 0 0 5.22E-06 8 94.64 0 0 0 0 0 1.86E-06 8 94.64 0 0 0 0 0 1.93E-06 0.38 8 94.74 0 0 0 0 0 1.04E-06 8 94.74 0 0 0 0 0 3.69E-07 0.39 8 95.00 0 0 0 0 0 1.13E-05 8 95.00 0 0 0 0 0 9.91E-06 0.44 8 95.11 0 0 0 0 0 2.33E-05 8 95.11 0 0 0 0 0 2.34E-05 0.48 8 95.23 0 0 0 0 0 1.07E-05 0.42 8 95.23 0 0 0 0 0 1.07E-05 8 95.34 0 0 0 0 0 8.56E-06 8 95.34 0 0 0 0 0 9.38E-06 0.41 8 95.95 0 0 0 0 0 8.57E-07 8 95.95 0 0 0 0 0 1.00E-06 0.39 8 96.22 0 0 1 0 0 0 3.81E-06 0.42 8 96.22 0 0 1 0 0 0 3.78E-06

8 96.32 8 96.32 8 96.45 8 96.45 8 96.57 8 96.57 9 93.36 9 93.47 9 93.57 9 93.67 9 93.77 9 93.93 9 94.14 9 94.26 9 94.38 9 94.38 9 94.53 9 94.53 9 94.63 9 94.63 9 94.73 9 94.73 9 94.82 9 94.82 9 94.91 9 94.91 9 95.00 9 95.00 9 95.20 9 95.20 9 95.30 9 95.30 9 95.40 9 95.40 9 95.50 9 95.50 9 95.61 9 95.61 9 95.91 9 95.91 9 96.00 9 96.00 9 96.11 9 96.11 9 96.22

0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0.8 0 0 0.1 0.9 0 0.8 0 0 0.0 0.2 0 0.8 0 0 0.2 0 0 0.3 0 0 0.1 0 0 0.8 0 0 0.8 0 0 0.1 0 0 0.1 0 0 0.1 0 0 0.1 0 0 0.1 0 0 0.1 0 0 0.3 0 0 0.3 0 0 0 0 0 0 0 0 0.1 0.3 0 0.1 0.3

0.39 0 0 0.39 0 0

0 0.1 0 0 0.1 0 0 0.2 0 0 0.2 0 0 0 0 0 0 0 0 0.1 0.3 0 0.1 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0.8 0 0 0.8 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.2 0 0 0 0 0

0.2 0 0 0.8 0 0 0.2 0 0 0.8 0 0 0.7 0 0 0.9 0 0 0.3 0 0 0.3 0 0 0.9 0 0 0.9 0 0 0.9 0 0 0.9 0 0 0.9 0 0 0.9 0 0 0.7 0 0 0.7 0 0

1 0 0 1 0 0

0.7 0 0 0.7 0 0 0.4 0 0.2 0.4 0 0.2 0.7 0 0.2 0.7 0 0.2 0.7 0 0.2 0.7 0 0.2

1 0 0 1 0 0

0.7 0 0 0.7 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0.2 0 0 0.2 0 0 0

1.01E-06 0.43 1.08E-06 4.00E-06 4.02E-06 0.43 1.03E-05 1.02E-05 0.41 7.52E-07 0.38 2.48E-06 0.36 4.10E-07 0.39 6.08E-07 0.36 6.24E-06 0.38 2.15E-07 0.34 1.57E-07 0.42 2.39E-07 0.37 7.20E-07 0.35 3.68E-07 2.52E-07 0.41 2.01E-07 6.90E-08 6.48E-08 0.66 7.10E-08 8.38E-08 0.32 2.81E-07 0.39 2.91E-07 1.98E-06 2.09E-06 0.36 2.19E-07 2.26E-07 0.65 5.26E-07 3.47E-07 0.34 8.42E-08 1.07E-07 0.38 1.49E-06 1.46E-06 0.41 6.78E-09 8.04E-09 0.55 4.96E-06 6.77E-06 0.40 4.40E-06 0.34 4.68E-06 6.21E-06 4.25E-06 0.31 4.49E-06 4.27E-06 0.38 7.45E-06

294 9 96.22 0 1 0 0 0 0 1.00E-05 0.41

93.18 0 0 0 0 0 1 1.01E-06 0.22 93.18 0 0 0 0 0 1 9.10E-07 93.27 1 0 0 0 0 0 2.01E-06 93.27 1 0 0 0 0 0 1.83E-06 0.32 93.36 0 0.5 0 0.5 0 0 5.59E-07 0.39 93.36 0 0.5 0 0.5 0 0 9.84E-07 93.46 0 0 0.7 0.3 0 0 1.26E-06 93.46 0 0 0.7 0.3 0 0 1.49E-06 93.57 0 0 0 1 0 0 2.00E-06 0.44 93.57 0 0 0 1 0 0 1.51E-06 0.19 93.86 0 0 0 0.5 0.5 0 4.81E-08 93.86 0 0 0 0.5 0.5 0 7.81E-08 0.39 93.97 0 0.1 0 0.8 0.1 0 1.42E-07 93.97 0 0.1 0 0.8 0.1 0 1.36E-07 0.42 94.07 0 0.0 0 0 1 0 3.76E-08 94.07 0 0.0 0 0 1 0 4.15E-08 0.44 94.18 0 0.0 0 0 1 0 6.83E-07 94.18 0 0.0 0 0 1 0 6.77E-07 0.43 94.38 0 0.1 0 0.7 0.3 0 1.60E-07 94.38 0 0.1 0 0.7 0.3 0 1.72E-07 0.41 94.48 0 0.1 0 0.7 0 0.1 2.11E-07 0.40 94.48 0 0.1 0 0.7 0 0.1 1.55E-07 94.58 0 0 0 0.1 0.9 0 5.96E-06 0.32 94.58 0 0 0 0.1 0.9 0 1.98E-06 94.68 0 0 0 0 1 0 1.59E-07 0.36 94.68 0 0 0 0 1 0 1.30E-07 94.80 0 0 0 0 1 0 1.94E-05 0.49 94.80 0 0 0 0 1 0 2.55E-05 95.80 0 0 0 0 0 1 1.14E-07 0.30 95.80 0 0 0 0 0 1 1.22E-07 95.90 0 0.8 0.2 0 0 0 8.39E-07 95.90 0 0.8 0.2 0 0 0 8.12E-07 0.39 96.02 0 1 0 0 0 0 2.73E-06 0.44 96.02 0 1 0 0 0 0 3.10E-06

12 92.98 0 0.0 0.0 0.2 0.3 0.5 6.52E-08 0.36 12 92.98 0 0.0 0.0 0.2 0.3 0.5 6.66E-08 12 93.09 0 1 0 0 0 0 9.60E-06 0.48 12 93.09 0 1 0 0 0 0 1.00E-05 12 93.20 0 0.8 0.2 0 0 0 2.15E-07 0.42 12 93.20 0 0.8 0.2 0 0 0 1.63E-07 12 93.47 0 0.1 0 0.9 0 0 4.95E-08 0.42 12 93.58 0 0.1 0 0.9 0 0 6.02E-07 0.41 12 93.69 0 0.2 0.5 0.3 0 0 6.30E-07 0.33 12 93.81 0 1 0 0 0 0 8.04E-06 0.38

12 94.01

12 94.01

12 94.12

12 94.12

12 94.22

12 94.22

12 94.22

12 94.32

12 94.32

12 94.42

12 94.42

12 94.62

12 94.73

12 94.83

12 94.93

12 95.03

12 95.52

12 95.64

12 95.64

13 92.17

13 92.27

13 92.42

13 92.48

13 92.58

13 92.74

13 92.85

13 92.96

13 93.07

13 93.18

13 93.29

13 93.39

13 93.49

13 93.60

13 93.70

13 93.82

13 94.06

13 94.17

13 94.29

13 94.40

13 94.69

13 94.69

13 94.80

13 94.80

13 94.91

13 94.91

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0.7 0 0 0 0.8 0 0.2 0.5 0 0 0 0 0.2 0.8 0 0.2 0 0 1 0 0 1 0 0 0.5 0 0 0 0 0 0 0 0 0.2 0 0 0 0.4 0 0.2 0 0 0.4 0 0 0.0 0 0 0.1 0 0 0 0 0 0.1 0 0 0.1 0 0 0.0 0 0 0 0 0 0.1 0 0 0 0 0 0.0 0.3

0.68 0.0 0 0 0.1 0 0 0 0 0 0.5 0 0 0.5 0 0 0.9 0 0 0.9 0 0 0.8 0 0 0.8 0

1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.3 0 0 0.1 0 0.1 0.3 0 0 0.2 0.8 0 0.0 0 0 0.8 0 0 0 0 0 0 0 0 0 0 0.5 0 1 0 0.8 0.2 0 0.8 0 0 0.6 0 0 0.8 0 0 0.6 0 0 0.7 0.2 0 0.9 0 0 1 0 0.0 0 0.9 0 0.9 0 0 0.9 0.0 0 1 0 0 0.2 0 0.7 0.9 0 0.1 0.2 0 0.4 0.3 0 0 0.9 0 0 0.3 0 0.7 0 0 0.5 0 0 0.5 0.1 0 0 0.1 0 0 0 0 0.3 0 0 0.3

295 1.31E-07 0.34 1.30E-07 1.35E-06 1.19E-06 0.49 8.40E-06 8.91E-06 8.28E-06 0.43 1.13E-05 1.19E-05 2.48E-05 0.35 2.34E-05 6.10E-08 0.39 1.97E-07 0.37 1.06E-06 0.35 1.84E-07 0.39 2.75E-07 0.34 4.57E-07 0.41 2.61E-06 0.43 2.67E-06

0.33 4.67E-08 0.37

3.52E-08 0.40 0.38 0.39

7.97E-07 0.37 4.59E-08 0.41 4.61E-07 0.40 3.40E-08 0.42 8.88E-08 0.42 1.47E-07 0.39 7.55E-08 0.45 9.17E-07 0.43 7.21E-08 0.34 6.60E-07 0.43 1.15E-08 0.34 1.29E-08 0.29 7.05E-09 2.72E-07 0.33 4.34E-07 0.32 4.46E-07 1.72E-07 0.37 3.36E-07 2.80E-07 3.53E-07 0.35

13 95.02 1 0 0 13 95.02 1 0 0 13 95.54 0 0.9 0 13 95.65 0 1 0

296 0 0 0 9.28E-07 0.35 0 0 0 9.84E-07 0.1 0 0 1.24E-06 0.53 0 0 0 6.07E-06 0.38

1111 111

Geosystem modeling with Markov chains and simulated annealing

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